I found Sunday's NYT puzzle by Ian Livengood entitled "A Bright Idea" a good challenge. Within the theme, there is a hidden message that spells out "AHA MOMENT", which is an aptly meta-level comment on the puzzle, and the psychology of puzzle-solving.
Researchers on problem solving have delineated a special kind of problem solving in contrast to the normal step-by-step iterative sort: insight-based. Bill Batchelder recently wrote an academic article bemoaning the state of cognitive theory on insight-based problems, and included 19 of these any puzzle solver will get a kick out of if they haven't seen them already. This includes the legendary 9-dots problem that makes you think 'outside the box'.
Normal "problems" you solve are often just daily chores or tasks. The act of solving an entire crossword is a normal problem; you have a general strategy which you execute, piece-by-piece, until you succeed. Both of these are usually well-defined problems in that they have a clear end-state (as opposed to ill-defined problems, which are much harder and include so-called 'black swans' problems). But insight problems tend not to be solved the way non-insight problems are. Insight problems are characterized by the so-called "AHA Moment" (which is now part of the scientific terminology).
The basic theory of insight problems handed down from Gestalt psychologists from the middle of the last century suggests that you can't deliberately solve them. You can work on them over time, and you don't feel like you have made any progress, even up until immediately before you get the answer. Then, AHA, the answer comes. This is the essence of solving individual answers of crossword puzzles. When a clue is easy, you see it immediately. When it is difficult, you search and search, never getting any closer, until AHA, the answer is there and it seems obvious in retrospect.
Theories of how to solve insight problems provide no help for speed-solvers. The best science has come up with is "Incubate". If you try thinking about something else, take a nap, come back a day later, often you will solve the problem immediately. Segal showed that incubation of 4-12 minutes improved solutions to insight problems. So, it might be worthwhile to put the puzzle down, or at least work on some other part of the grid, returning five minutes later. Of course, the top solvers will have solved 2-3 additional puzzles by then, but maybe those folks don't easily get stuck.
The other thing about crosswords is that because they combine insight and step-based problems, a difficult insight problem adapts as the puzzle gets solved, and becomes easier as more letters are provided. This makes it easier to access via the letter-clue route, making it more like a non-insight problem.
In terms of A Bright Idea, I experienced several insight problems. For example:
25D Part of a moving line _ _ _
This one was tough for me; I first thought of CONGA (doesn't fit), which threw me off on 22A Venemous Tree Dweller (which I mistakenly answered SAMBA instead of MAMBA). A moving line could be a line outside a concert or ballgame, and once I got _ A N, MAN and FAN seemed possible, but not very good. Even when I had _AN, the correct answer never came to me, and I got it by figuring out the crossing V; of course VAN makes sense; a moving line also refers to an outfit like Hertz or UHaul. BAH.