I have had a CTX100 Toshiba phone system for at least a decade. All of a sudden I am sometimes unable to answer a ringing phone. The phone system thinks I have picked up the phone, but the call continues to ring. Any ideas?

what type of phone you have? i have noticed in the past with old DKT 3020 or 3010, the hook switch is bad and do the same problem.
check replacing the phone with a new one DP5000 series & try.

I know you've likely already checked this, but I've found that the line cord (wall to phone) being bad will sometimes do this. Had one just yesterday, as a matter of fact. (They're rather abusive to the phones around here, pull, yank, throw)

With a system this old, and Toshiba out of business, problems like this will only get worse. Time to think of moving to today's Technology.

We are starting to talk to our Toshiba base about moving forward.

Could have been as simple as a hook switch in the older style phone.

These system will keep running for years to come so there will be plenty of sales scare mongering of having to move on quickly.
Don't forget that the Toshiba has a 5-7 year warranty depending on the country.