The weakness Mute often means that a character cannot speak. Here's a variation that allows the character to talk--but not too much.

Weakness Description:
Characters with this weakness pledged never to reveal important aspects of their powers or past. Maybe they were trained by an enemy organization. Or perhaps their powers derive from an extra-dimensional source that wants to remain unknown.

Whatever the case may be, the vow of silence must be significant enough to interfere with a character's personal life or interactions with others beyond the usual limitations of a secret identity.

Breaking the vow of silence (even due to mind control or torture) will make the character distrusted or demoralized, possibly changing this weakness to Prejudice or Reduced Charisma (GM's discretion).

Depending on the circumstances, the character may be prosecuted by authorities or hunted down by someone from their past if they reveal their secret to anyone--even in private.

I really like both of these, especially the Mute one.  Makes me think it might be nice to compile a list of "alternate uses" of some of the powers we tend to allow in our campaign.