This hand happened in a vegas casino where the rule is significant action = 3 checks.

3-way to flop in a limit game.

First guy doesn't act, second guy checks, I'm last to act, I think for about 10-15 seconds then decide to check.

Dealer burns card, then the first guy claims he never acted and now he wants to bet.

There wasn't significant action, but he waited over 15 seconds between the guy acting out of turn and saying he didn't act. Clearly if I bet he would have just checkraised, but instead decided to see if I bet and then object. I was the preflop raiser so he most likely would have checked to me anyway, and if he wanted to bet he would have already in the 15 seconds that passed.

So is there any rule that applies to this situation ? There wasn't significant action but there was a significant amount of time between the out of turn action and his objection.

I would probably not allow him to bet.

The amount of time that passed makes me think he was allowing the action to pass him, so he should lose his right to act even if the number of actions behind him doesn't quite meet their rule.

If there had been 3 out of turn insta-checks behind him, I'd be more likely to allow him to bet then than I would be here.

The rule that my poker room uses is three actions without money (3 checks) or two actions with money. We consider the dealer tapping the table as an action to avoid these situations.