Most of us somehow learn that I, he, and she are subjects and that me, him, and her are objects.  But put these pronouns in combination with another noun or pronoun (a compound), and many speakers lose their wits.  People who would never think of saying, "Me went to the movies," "Him went to the movies," or "Her went to the movies" will blurt out "Mary and me went to the movies," "Him and Mary went to the movies," or "Her and Mary went to the movies."

This makes no sense, for the presence of Mary in the sentence in no way alters the fact that I, he, and she are the subjects.  Similarly, people who would never think of saying, "Send your answer to I" will say, "Send your answer to Mr. Jones or I," even though it's quite obvious that the presence of Mr. Jones does not alter the positioning of the personal pronoun as the object of the preposition (so the correct pronoun is, of course, me).

Unsure of themselves on this point, some take refuge in "Send your answer to Mr. Jones or myself," which is equally absurd because one would never say, "Send it to myself."  I become annoyed by these obvious solecisms because one does not need a grammarian's knowledge of pronoun case to know that they are wrong; one needs only to use one's ear and a touch of logic.  However, some people cannot open their mouths without inserting an ungrammatical foot.  Next time you hear one of these statements, dare to be politically incorrect and point it out.