Is it by any means possible that if you update the kernel on a large system, it will cause any critical failures?
It probably won't break anything right away, but any number of things could go wrong when you reboot and try to load the new kernel:
the kernel may not have been loaded completely (e.g. if you've run out of disk space on your
/bootpartition or something went wrong during the update): system won't boot
you might have installed an incorrect kernel for your hardware (especially if a previous administrator had installed a custom kernel): system won't boot
the kernel might not be compatible with your applications or missing some required features or compiled-in modules (especially if a previous administrator had compiled and installed a custom kernel): applications won't run or won't work correctly
there might be new bugs in the new kernel: applications won't work correctly
Most of the time, everything will "just work" and you'll be fine. The closer you are to a normal well-tested distribution, common hardware and often-used services the less likely anything will go wrong. But whenever you change something, there's a possibility that something will break, either loudly and immediately or subtly in the future.
If you're running critical services you should test the upgrade (including all the services after completing the upgrade) on a similar piece of hardware before upgrading your actual hardware, you should perform the upgrade when the critical services aren't in demand or have a failover system take over during the upgrade, and you should have a (tested!) backup/restore procedure and enough knowledge to fix any problems you occur before they interrupt your services.