I saw the Jickets open for They Might Be Giants at a club called "Neither Nor" on the LES sometime in the early mid 80's. They were pretty hilarious. They were kind of a teeny bopper band parody, in fact I came across a xerox-ed "fanzine" they had made about themselves in a Rivington St. bookstore years later. Most of their songs have the word "Jicket" in them (I love when bands do that self referential thing...). My old roommate Mark Spencer is an even bigger Jickets fan than me and actually covered The Manhattan Bridge song with vocals (the version here is instrumental). This came out on the Hello Recording Club label which has something to do with They Might Be Giants.
UPDATE: Drummer Petey comments: "Nice of you to mention the Jickets. However, the only Jickets song that mentions "Jickets" is in fact "Jicket Island!"