Well, in that case, I suppose my question is "what are some examples?"
Can KPPP hold a draw against KPPPP if they are on the same side?
@NervesofButter - 1.Kd2 Fine's advice: Begin by advancing the king as far as possible.
All eleven of white's available first moves suffice to win. 1.e4 and 1.Kd2 have the most to recommend them.

If it's down to just kings and connected pawns on the same side of the board, is it ever possible for the side with 3 pawns to hold a draw?
Ever possible? Yes in some positions. Usually? Usually, the 4 pawns versus 3 pawns should still be able to convert the endgame to a win with best play (or side with extra pawn).
If you want more information of what types of pawn formations are better able to hold against other pawn formations, then the Hans Kmoch book, Pawn Power In Chess is a book I recommend. The only drawback is that I don't think this book was ever translated into algebraic chess notation, so you'll have to learn English Standard/Descriptive notation to read it.
https://en.wikipedia.org/wiki/Descriptive_notation
If it's down to just kings and connected pawns on the same side of the board, is it ever possible for the side with 3 pawns to hold a draw?