As above, if there is even a remote chance you are interested, just pick a date so that you are on the schedule. It is much easier to change dates (albeit not easy) than to get an interview if you wait and all the spots are taken (and you have to wait for someone to drop out).
There are plenty of anecdotes here of people waiting to arrange travel, etc. or see if another program was going to invite them, only to find out that there were no more positions for interview open when they finally called.
I would call/email the SAME day I got the invite and to be honest, in a very competitive field, I would be checking my email constantly, so that I was not more than a few hours out from an invite.