Depends on what you mean by "a little bit". If you did this like an hour ago, and the worst thing you're feeling is wondering whether you'll be alright or not, then you will probably survive. Still, I have to ask: How did you manage to swallow a little bit of nail polish remover?
It probably will not harm you if you swallowed a little bit but it all depends on the amount of remover you ingested. The acetone in nail polish remover might dissolve tissues in you throat and stomach if taken in large quantities. It would be painful and in one case a person developed diabetes.