Turns out they aren't getting DHCP addresses after all. I checked both the dvswitch and our DHCP server and both are only at 60 - 70% capacity currently and maxing out at 75%.
While running the support tool I realized that the network connections were had the little caution flag on them (Windows 7 desktops). So I guess the new question is how can I make sure that these machines always get a DHCP address when they get provisioned?
Edit: Although I don't think this is a VMware issue now, do you have any thoughts as to what I could do to make sure they are getting DHCP addresses?