Any such dimensions cannot be infinite like the three spatial dimensions we're familiar with, otherwise we'd have noticed because gravity would be orders of magnitude weaker, especially if there's 11 infinite dimensions. These other dimensions must be closed and very very small for us to have not noticed yet, since gravity appears to precisely obey inverse square law. It's a question of measurement.

I would hope it's not the only prediction string theory makes because otherwise if it's wrong we'll be waiting a long time to confirm it's wrong.