How about when the updated browser removes functionality that servers require?

Especially in 'enterprisey' networks there are devices that /still/ require SSL, Java applets, Flash (urghh!), SSH v1 support, or features of (X)HTML, CSS, and Javascript that have been removed or modified. Most of that hardware would be expensive and disruptive to replace.

I still maintain a version of 32-bit Netscape Navigator for accessing some key infra-structure devices that continue to work perfectly and are on isolated private networks.

Some examples: Cabinet Distribution Units (CDUs - network-controlled power switches), network KVM (Keyboard-Video-Mouse over Ethernet), Ethernet switches