I don't know if its "cool" but its certainly possible. I've had guys break up with me and instantly start dating another chick. With that said.... From the ex's prespective it is quite hurtful. Sorta makes one believe that the loss of their reltionship with you isn't worth a greiving process. Besides I think someone should take time to heal from a break-up before jumping into a relationship.... But that is just me.
Theres is also the scenario where a person hooks up with someone that they've been wanting to date even while they were in the previous relationship (that may even be the reason for the break up) In that scenario the fast hook up is more understandable... but still hurtful to the ex.
Either way I wouldn't say you're "in between relationships" if you've already comitted to another person.... then you're IN a new relationship.