Question: Can you mate with just a Knight + Bishop?

Hmm, ok.

I would disagree with much of that.

It is generally unnecessary to move the knight and sometimes also the bishop, but if the intention is to exclude the lone king from the centre a knight in the centre is not very effective. Also if the lone king is played correctly (as opposed to accurately) the lone king will refuse to go to the wrong corner from the great majority of positions. A computer program with an attached EGTB will almost always move his king to the wrong corner in preference to abandoning the wrong diagonal, but the EGTBs do not play the lone king in this endgame correctly. See my post #116 in https://www.chess.com/forum/view/endgames/bishop-knight-amp-king-versus-king  

If the player with the knights wants the quickest mate he should look to permanently exclude the lone king from the wrong diagonal at the earliest opportunity rather than actively seeking to manoeuvre the king to the wrong corner. This can be done from the middle of the wrong diagonal in some cases.   

It should be possible to mate with a bishop and a knight. Might take a few moves to get the king into a corner. It is more obvious (forthcoming) with a rook. Best to ask the opponent to kindly resign. :-)

I don't try to make the quickest mate, but I chose a mate that makes sense for me and is therefore easier (for me). I learned two  different systems, but know I tried only the method showed in the book "100 Endgames You Must Know". You can train it by spaced repetition in Chessable. The method described is rather straightforward:

"1 ) Transferring our king to the centre, to drive the black king off.

2) Transferring the knight to the centre as well, since it is a short-range piece.

3 ) Pushing the black king to the edge, where he will head for a corner opposite to the bishop's colour (otherwise, checkmate comes sooner). We will call this corner a Safe Corner. Here, it is the a8-square.

4 ) Our king occupies a square on the long diagonal opposite to the corner (here, the c6-square). We will call this spot the Pivotal Square.

5 ) The knight drives the black king off the Safe Corner from c7 or b6.

6 ) The bishop drives the black king off the square adjacent to the corner from a7 or b8.

7 ) We force the black king to a corner of the bishop's colour (Mating Corner) by means of an accurate manoeuvre. We will give more details when the moment comes.

8 ) The black king is locked in the cage. (you have to buy the book to see what the author means with this).

9 ) The white king stays a knight's jump away from the corner (f7 or g6). We will call this spot the Mating Square.

10 ) We arrange the checkmate, which will come by two consecutive checks, one with each piece.

The execution of Step 7 requires special attention."

As I said, I prefer to learn the whole thing with spaced repetition, so I will memorize all the steps and review them from time to time.

How about on a 12x12 board (or even higher dimensions)?

Why?

Capablanca's board is 10 x 10, perhaps the future of chess. Funny that in this variant check mate is easier and quicker - we could expect the opposite of it.

Martin, you don't play games in chess.com? Why do you have such a low rating?

As mentioned in my previous post, if Black plays correctly you probably won't arrive at the position you show. If you play against an EGTB in most cases it is not difficult.

The main point is to walk the White king to f6 up the wrong diagonal and adjacent diagonals of the opposite colour while the Black king is not in the rectangle with corners a1 and the White king's square. square (this could be the opposite corner depending on the king positions). The Black king must walk back to the h8 corner or be sealed behind one of the diagonals adjacent to the wrong diagonal when mate is normally quicker with accurate play. The EGTB will therefore normally walk back to the wrong corner. The Black king can oppose from the front but the opposition can simply be broken by attacking one of the squares it needs with a piece (there's no rush - the Black king's not going anywhere). The only real complication arising is a possible attack on the knight allowing the Black king to leak back into the White king's rectangle, so if you want to be lazy move the knight out of the way at the outset and leave it there, that way it also shouldn't get in the way of the other pieces.

Once the White king has reached f6 the White king should be restricted to h8 and the adjacent squares g8,h7 as below.

Should the Black king venture off the three squares mentioned then the bishop can seal the king behind the appropriate diagonal. E.g. 1...Kf8 2.Bd5 with Black to play. (There are some stalemate cases where the king can venture one square further without any significant difference.) Having reached a position similar to the above and with the Black king trapped you can now just move your pieces to the position you're aiming for. 

Probably because I don't play games in chess.com.

Why don't you play games on chess.com? Chess.com is the BEST.

Because I'm a rubbish chess player. Look at my rating.

Your rating is 800 because you haven't played...

It's a reasonable overview as these things go but step 2 I think is misguided as mentioned previously and step 3 begs a few questions such as, "What if he doesn't want to go?" and "What if he heads back out again instead of heading for the wrong corner?".

I had intended some changes to the Wikipedia page for this and looked at several possible sources. The great majority of positions are what I've called "open" positions where the lone king can reach the wrong diagonal but neither in the manuals nor games could I find any extended accurate play in these positions, nor anything but the kind of generality in 3. Most agree that you should use all your pieces to drive the king to the edge or sometimes wrong corner, but it doesn't translate into accurate examples. I usually don't.

I worked out my own (mindless) method when I first learned the ending which works from any won position but is not necessarily optimal. I would recommend that approach rather than reading.

These days I practice against the Nalimov EGTBs with the aim of reaching the DTM (as either colour). It's not necessary to play the ending but I'm hoping to come up with a similar mindless method of mating optimally this year. 

By the way if you're not interested in optimality you can also dispense with 7. From the wrong corner you can reach Delétang's first net in a few moves, so if you want to learn a method that would probably be more useful. Philidor's method and similar are only methods of driving the king from a wrong to a right corner. 

Mate on a 10x10 board is typically neither easier nor quicker (and less often possible I believe). Mate on a 9x9 board is easier so long as your bishop is the colour of the corners, because there are mating corners in all directions and the lone king is never more than four moves away from one.  

How is your rating? Play some games and you soon will find it out...

Actually I just learn this mate because chess culture, and also perhaps to learn more about this specific coordination of pieces. I only played twice this ending. The first time with king alone, and my opponent said it is a forced draw, I said he is wrong, so I took his side and won (it was easy for me, as he went first to the wrong corner, and was trying always to go back there, who make things easy for me). The second time I was again with the king alone, but the guy was not able to mate in 50 in a blitz. I didn't spend much time working out the details of this ending, and I just hope to be able to mate when I have the opportunity after learning the system proposed in the 100 endgames-book.

Therefore, we can say that MARattigan is undefeated in all time controls and game formats on Chess.com so far.

Just purely out of curiosity.

This is one of things I'm determined to have competence in 50 moves and less.