What is the limit when applying for the same internship that you did not get the first time?

Though it might be discouraging to miss an opportunity the first time, I encourage you to apply at least once more. However, in-between applications, take a look at the skills and experience necessary to be successful in the role, and work on attaining these aspects prior to submitting your application. If possible, have an industry professional or career counselor take a look at the internship requirements compared to your resume for some feedback into reasons you may not have initially been selected. A second set of eyes and fresh perspective couldn't hurt when trying to land that dream internship. Good luck to you!

Your only limited by your resolve... keep trying, if they get sick you they'll let you know.

I'd say go for it as many times as you want. Make sure you have something new to bring to the table each time though. Be ready to explain why your more qualified now than you were the last time you applied.

I'm going to be slightly different than the advice received above and suggest that too many applications might be a deterrent. You certainly don't want to close any doors for future internships or even employment with a company that you think you might like to work for. Several might be ok if you have different things to add or a new position to explain, but more than just submitting multiple times, I might try a different approach and see if you can figure out how you (or someone you know very well) is connected to the company and can help you make a personal connection and even put in a good word for you to pull your application to the top of the pile. LinkedIn is a great way to see where people know people, too. It's these relationships that often make more of a difference than anything else. Not everyone who gets jobs "knows someone," but it sure helps at least get your foot in the door.