Can I apply a second time for a J-1 visa?

You can now apply for the J-1 visa categories INTERN and TRAINEE numerous times.  Note the following regulations:

INTERN:

Repeat application are possible as long as you are enrolled in a post secondary institution and seeking a degree.

TRAINEE:

There is a 2 year homestay requirement prior to being able to apply for the J-1 Trainee category.  This also holds true if you previously held a J-1 Intern visa and are now applying for the J-1 Trainee visa.