I want to know whether expressions such as お預かりさせていただく and ご説明させていただく are considered 二重敬語. お・ご～する is one way of making a verb into 謙譲語, and ～させていただく is another. Both are quite common, and I hear the combined form frequently as well; however, searches have provided conflicting answers. I am leaning towards that it is not 二重敬語, but even if it is it's generally accepted. I hear it quite often in official announcements, after all.
They aren't 二重敬語 (but 敬語連結) but I'm thinking the problem if they sound wrong or not is parallel to that of 二重敬語. Whether it's 二重敬語 or 敬語連結, the problem is if they sound redundant, indecent in a certain situation or so. It's not a problem of grammar.
Personally, I don't like recent tendency to abuse させていただく even when the speaker is supposed to be responsible for the action.
Edit: I'd like to clear out possible confusion. First, 二重敬語 is a grammatically correct expression. It just could be over-polite and indecent. After WW2, some people claimed that it's politically incorrect to use in public speech because it had been formally applied to imperial family member. (In private speech, it would be just randomly used.)