In almost all cases, it's fine. A few contests might specify that they do not accept simultaneous submissions, but I have found these to be the exception. (And they will spell it out clearly in the guidelines.)
If you do win a contest or the piece is accepted for publication elsewhere, you will obviously have to withdraw from all other contests -- and that means forfeiting your entry fee. So as long as you are prepared for that possibility, it's fine.
If you're going to enter contests, and if you have the means to do so, I think entering more than one at once is a good idea. Besides, if you win the $1,000 prize + publication, who cares if you lose your entry fee at a few other contests? Good luck!