Please have a look at the configuration maximums guide for vSpehre 5.5.
http://www.vmware.com/pdf/vsphere5/r55/vsphere-55-configuration-maximums.pdf
What you are looking for can be found in the section "vCenter Server Maximums"
Concurrent vSphere Client connections to vCenter Server 100
Concurrent vSphere Web Clients connections to vCenter Server 180
Please note that these are the tested values, you can experience more or less connections depending on what is actually done in those connections.