I'm trying to recall a particular chess problem. It's related to retrograde analysis. All what I can remember is this: it's an endgame where there is only one way to reach the position, and during last 49 moves there were no captures, so a potential winning move draws instead.
There are dozens of problems that illustrate a potential winning moves that instead leads to a draw because of the 50 move rule.
One example is the following mate in four published by Léon Loewenton in 1956 :
[fen "5KBN/p2ppp1r/1p4pp/b7/RP6/1PP4P/1RpPPPkP/n1B1Q1N1 w - - 0 1"] 1. Nf3 Rg7! 2. Kxg7! (2. Qg1+?? draws)
There is an apparent mate in three moves: 1. Nf3 followed by 2. Qg1(+) and 3. Qg3 mate.
However, after 1. Nf3 Rg7! 2. Qg1+ black claims a draw according to the 50 moves rule (the retrograde analysis is left as an exercise to the reader!).
If white plays 2. Kxg7!, the capture breaks the 50 move chain and white can mate with 3. Qg1(+) followed by 4. Qg3 mate.
Welcome to stackexchange.com, dveim.
There's a great answer already, but I think there's a couple of interesting points that I can add.
First, in the composition world, there's a convention which states whether the 50 move rule applies by default only to retro-problems. See Chess Problem Codex Article 17. So all these very long "cursed wins" found in the tablebases don't get interrupted. However the retro problems mostly rely on the rule. But for some problems the convention would work the wrong way round, and in these cases the convention would be ignored, e.g. Elkies' famous problem.
Now, some problem composers think that castling should reset the move count. I personally think that this is illogical: if this were so then any king or rook move which disrupts castling rights should reset the move count. But there are great enthusiasts for this interpretation.
Another corner-case is what happens when the 50th move delivers checkmate? The FIDE rules for 75-move draw are clear: mate trumps draw. However to conclude the same thing for 50-move rule requires looking at the mechanism for claiming draws. It assumes that no-one delivering checkmate would want to claim a draw instead, and that coupled with the fact that someone would be checkmated before they can claim a draw in their own turn effectively means that "mate trumps draw" for 50-move rule. But, although this is the FIDE intent, it doesn't explicitly say so, and some composers like the idea of creative ambiguity here. I personally think that mate trumps draw for 50-move rule.
Finally, there are currently 160 problems in PDB Chess Problem Database See overview here, of which an amazingly proportion of 94 are by the late great Russian composer, Nikita Plaksin.
EDIT: STOP PRESS: I deliberately skipped over the fact that the Codex of Chess Problem Composition had no formal notion of 50-move rule (or draw by repetition)! The sections of the FIDE Laws which cover these were deemed not relevant by the Codex. In the last few days, in Vilnius, at the annual Congress for Problemists, Footnote 12 of the Codex has been fixed in this regard. There is still the problem of how to translate these rules to the Problem world - the castling and checkmate issues are problems for the 50 move world, and the happy acceptance that composers are “free spirits” means you can end up with vague compromise assertions which are a barrier to retrograde analysis.