On Tuesday, they actually did sign a fourth quarterback.
Former Pittsburgh QB Bill Stull returns to Kansas City, according to Josh Looney of KCChiefs.com. He worked out for the Chiefs in their rookie minicamp in May. Since it was a rookie/first year camp, Stull was the only guy taking snaps so the Chiefs have seen plenty of him.
To make room for him, the Chiefs released former K-State DE Jeffrey Fitzgerald.
Stull's arrival is likely due to the injury suffered by Brodie Croyle. It doesn't appear serious because he practiced some on Sunday but on Monday he was seen with a brace on his arm. My guess would be that after this week the Chiefs will make a decision as to whether to carry four quarterbacks.
Generally teams need three quarterbacks in practice so maybe this is an admission that Croyle will be out for a few days.