The simple answer to your question is, no. Convictions do not automatically drop off your record after x number of years the way credit problems do. They are with you forever.
You can see about having the record expunged, but even that is no guarantee that it will never show up on a thorough background check.
Without knowing anything more about the situation than I do, understanding that I could be entirely wrong and absolutely meaning no offense, I'm going to take a couple of guesses and offer some advice based on those guesses. If my guesses are wrong, I apologize in advance. You are free to take the advice or not as you please. It applies even if I am wrong.
I am guessing that you assumed that (a) employers are prohibited from going back further than (it's usually 7 but sometimes 10) years or (b) you assumed that after x amount of time (again usually 7 years) the conviction just went away. Based on whichever of those assumptions you held, you either neglected to mention or out and out lied about (probably the first) the conviction either in the interview or on the application. You lost the job not because of the conviction, but because you failed to disclose it.
(I could be wrong. There could be something about the particular job you were looking for that made the conviction a factor. But most employers will not care about a conviction for a bounced check. Everyone's bounced a check or two on occasion.)
My advice is that in future, you treat any questions that relate to this misdeamenor quite literally. In answering such questions, many people assume that "have you ever..." means "have you, in the last 5-7-10-whatever years". It does not. Ever means ever. If the employer means 7 years, they will say 7 years.
You do not need to volunteer any information that is not asked for. But you should treat questions about convictions absolutely literally. Do not assume that it means anything other than it says on the surface. If the question specifically says, within the last 5 years, and your conviction was 7 years ago, you can truthfully answer no. But if it says, ever, that means ever. This is the point that most people get themselves into trouble over.
Again, if I'm guessing wrong, I apologize, and I mean no offense, just helpful advice from one who has conducted many a reference check and hired many people. Good luck to you.