[vuroth]

I opened 1NT, passed out. LHO leads the ♠4:

Scoring: MP

4♠ 2♠ T♠

Outside entries to dummy doubtful or non-existant.

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I couldn't decide if playing the J or the A was better, and I rushed my play rather than think it through.

If the J is better here, are there ever times where the A is better?

[Fluffy]

you really need the entry for something useful.

Think this way, if you win the ace you will paly low to the queen and get your entry, but at the same time you stablish the suit for LHO.

If you have a side suit such as ♦AK32 opposite ♦75 you might not care to stablish tricks to opponents, so there will be cases for it, but not many.

[Phil]

It depends when you need the entry. If immediately, you could have tried the ♠Q. Surely this isn't any worse than winning the Ace.

If you would like a delayed entry (there's a source of tricks over there), then winning the Ace makes sense.

[vuroth]

Mostly, I was hoping for 3 ♠ tricks.

The full hand was:

Scoring: MP

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So I need a 7th trick from ♠ or ♦, and in ♦ I think I'm happier if they break the suit.

Not really a puzzle hand, and maybe not even an interesting hand, it just occured to me at the table that there might be times when taking the ace would be clearly better than playing the Jack.

[vuroth]

Ok it took me a while to puzzle it out, but here's my thoughts.

Spoiler

It's at least plausible that RHO played the T from KT. If he did, my only shot at 3 spade tricks is to play the J at trick one.

This seems like the best play. IF LHO has the K, I can play J, then A, then low towards the Q. RHO must rise with the K at trick 3. I get a 4th spade trick if LHO has the A♦, or if RHO breaks ♦ for me.

A, then J never costs wehn LHO has the K, but does not help, except in one extremely specific case: LHO erroneously covers the J with the K, and the ♦ entry never materializes, AND RHO starts with the T8 exactly, giving me a cold finesse of the 9 for my third ♠ trick. Counting on LHO to make a ridiculous play is of course bad bridge, but sometimes bad bridge is rewarded. They key is to recognize that this is not a valid line.

Obviously, this was the line I took ATT, and it worked for me. Still, it was the wrong line, and the worst thing I could have done was walk away thinking I'd made some clever play.

I think a better example might have been

Scoring: MP

4♠ 2♠ T♠ ?
No outside entry

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Here, Ace first costs me nothing, and my only hope for more than 2 ♠ tricks is LHO incorrectly covering the J.

[Old York]

The only "correct" time to win the Ace from AJx is as a deceptive play, hoping to trick LHO into continuing this suit and not switching to a more dangerous suit

Winning with the Ace when you hope to establish 3 tricks in this suit seems very optimistic, you were very lucky to find the suit breaking so favorably, leaving Q7 sitting over 96. Perhaps West was foolish to win trick 2

The overtrick was handed to you by the defenders switching to Diamonds and never attacking Clubs, a lucky 91.67%
If you examine the first 3 results on the traveller, you will see that defensive errors gave away an overtrick on all 3

According to GIB, your mistake was discarding a Club on the last Spade, but your oppo gave you a second chance by continuing Diamonds

Tony