An endgame of Rook and Bishop vs. Rook came up in one of my live chess games. I eventually won it by pushing his king back until my combined attack force overwhelmed him. If you can find any improvements for either side please post them

On move 16 white should play Rd1 and if you move your bishop he should check your king making it impossible for you to progress in that position. White also makes the mistake of moving his king away from the rook and moving towards a corner when nothing is forcing him to do it.

Apparently your opponent didn't know that if he trades rooks, it's a draw.

It's a draw, but not a basic draw at all! The defense is difficult if not known. In Reuben Fine's Basic Chess Endings, he spends 6 pages going over this position. On contrast only a page is devoted to B+N+K vs K

White's mistake seems to have been outcasting his rook from the defense with 10. Re8. From there he seems doomed. 

Maybe because the above situation is common and B+N+K vs K never happens in real games?

You are probably right that he spends more time on this because it is more "common" but even at that it is still a very complex ending where the weak side can go astray and find himself lost. Six pages of analysis isn't light stuff!

I got it in one of my first-ever USCF tournament games.  I was only able to defend for about 25-30 moves.

White's first error occurred at 10 Re8. Necessary was 10 Re3. After 10...Rd2+ 11 Kd1, B has a win with 11...Rd3 12 Rb8+ Bb4 13 Rb7 Rd8 14 Rb5 Rc8+ 15 Kd1 Re8! with unstoppable mate at e1 unless W takes the B (which of course leads to K&R mate). However, B gave the draw back with 11...Ra2. W could now draw with 12 Rb8+, but played 12 Kd1, and B again has a win with 12...Rd2+ (rather than the drawing 12...Kc4 actually played, when W should now play 13 Re2, which draws). W tosses the draw again with 13 Rc8+ and B finds the right line this time with 13...Kd3 14 Rd8+ Bd4. However, I'm not sure why W resigned here, since there is a lot of play left and B could easily err: 15 Rc8 Rf2 16 Re8 Bf6 17 Re6 Bc3 18 Rd6+ Bd4 19 Re6 Rd2+ 20 Ke1 Ra2 21 Kf1 Rf2+ 22 Ke1 Rf5 23 Rd6 Rg5 and W cannot stop mate at g1 without giving up the R.

What a pain for the player who is down!

Black missed winning White's Rook twice with ....Re1+ and ....Rxe8.

The first time was with 10...Rc1+ (Instead of 10.Rd2+)11.Ke2 Re1+ followed by 12....Rxe8 and wins

The second time was with 12....Ra1+ (instead of 12...Kc4) 13.Ke2 Re8+ followed by 14....Rxe1 winning.

1.Ra6 instead of 1.Rg2 was best. Black's K is cut off from the rest of the board and would not be able to support a mating attack on White's K.

White had a second opportunity to cut off Black's K from the rest of the board after 7...Kb3 with 8.Rc8.

It is basic endgame technique for the defender to keep the K in the center of the board as long as he can and make it very difficult for the opponent to drive the K to the edge of the board.

Black can easily make progress by using his bishop to block white's rook and move his pieces up the board.