From my experience only one request is allowed in the queue at a time. So if someone has a request in the queue it will not allow anybody else to make a request.
Yea as soon as the requested song plays and clears out of the queue it will allow another request, so that request probably played before you made that second attempt.I ran into that yesterday using Chrome. It showed one request in queue by another user at the time. I tried again using IE11, and the request went through, but I didn't pay attention as to whether the queue still showed a request from another user on the IE attempt...so it may have just been the timing of the request and not the browser change that let it go through.