One of the guest machines was reporting an error when I tried to power on the most recent snapshot. Snapshots further back in the timeline powered on without any problem.

Error: Unable to change virtual machine power state: Cannot find a valid peer process to connect to.

Apparently I’m not the only one with this problem. Others have been reporting it since at least early 2005. The forums say to delete unused lock files and restart any hung VMware processes (or restart the host machine), which did not help.

Then I tried to clone the snapshot before making any more changes – just to have a backup – and that failed too!

Cloning failed: reached the host's limit for open files

Thinking the two errors are related, I looked into how many files are actually part of this VM. The list was HUGE. (more than 1500 files in the folder)

I had 13 snapshots before the one giving me trouble, so I deleted a few of them to reduce the number of files VMWare would need to open.
This fixed everything.

I could have increased the max number of allowable files, but some of the snapshots were not needed any longer.  Your situation might be different, but this is a solution I couldn’t find anywhere else on the web. Hope that helps someone.