In the board position both knights can move to e4. To distinguish which knight moved one would normally write Nce4 or Nfe4. However in this special situation only Nce4 is check.
Would it be sufficient here to write Ne4+ (meaning Nce4) only (i.e. without the "c" but with "+"), as Nfe4 would not be check?
By sufficient I mean whether there are any (FIDE or other organization) laws regarding such issue.
8/8/5N2/8/3k4/2N5/8/B3K3 w - - 1 1
Would it be sufficient here to write Ne4+ (meaning Nce4) only (i.e. without the "c" but with "+"), as Nfe4 would not be check?
It depends on exactly what you mean by "sufficient". By "normal" definitions of "sufficient" the answer is obviously "yes". "Ne4+" disambiguates and leaves the human reader in no doubt which knight was moved.
However it doesn't work in FEN diagrams. You need to specify Nce4 to get it to work. So, then the answer would be "No".
The FIDE Laws of Chess describe how you should annotate your games in Appendix C and that does not allow for using check as a way of disambiguating between two different pieces of the same type moving to the same square. However there is no penalty specified for mistakes. It is up to the arbiter to decide and that will only happen if it is noticed.
In real life the most that might happen is that if the games are transcribed for sending to FIDE or for publishing the player might get a scolding from the arbiter tasked with this chore, but this is extremely unlikely in this case and would happen long after the game is completed.
Mistakes far worse than this happen all the time and the only comeback from the arbiters is when the scoresheet is completely illegible or grossly incomplete. Vigilant arbiters will notice these misdemeanors during the game and warn the player concerned to mend their ways.
The FIDE Laws of Chess 2018 are unambiguous: you should write Nce4, and the + sign is optional and not disambiguating. Ne4+ is not sufficient.
Appendix C (algebraic notation), article C.10 basically tells you to write Nce4 to distinguish it from Nfe4. It does not mention check or other ways to disambiguate.
Article C.13 notes that writing "+" to indicate check is optional, which also implies that it does not serve to distinguish between a Ne4 that checks and one that doesn't.
Thank you, Remellion, for providing a clear answer based upon the FIDE Laws. If the "+" is optional, it stands to reason that a PGN reader may presume this symbol may be discarded entirely, prior to processing moves of the game. Thus, it can not be correct to use the check symbol as a disambiguation, as it definitely results in ambiguity.
I presume checkmate and stalemate symbols are also optional. Thus, in this diagram:
[FEN "QQ6/Qp6/2k5/8/8/8/1R3RN1/3N2KB w - - 0 1"]
And for those still not convinced... think of it another way... Suppose you give a beginning chess student this diagram as part of a hand-written #1 test (note: if so, first remove all white units except wK and 3 wQs). Now, who would honestly accept "Qxb7#" as a solution? Nobody! This is a clear failure to disambiguate which queen delivered the mate. Zero credit.
