Most people insist that 'and I' is the correct form.

It's "I" if it's the subject of the verb (the do-er of the action) and "me" if it's the object of the verb (the done-to of the action).

Thus "My instructor and I think that..." and "It seemed to me and my instructor..." are both correct. (There is no grammatical reason to prefer "My instructor and I" to "I and my instructor" or vice versa, but the former is more usual.)

It's exactly the same distinction as between "he" and "him", which for some reason are far less often misused.

Peter