There is a heavy wire running from the outside phone box (company side)
to a big spike in the ground. I clipped that with hedge clippers some
years ago, then I repaired the break merely by twisting the ends back
together and that worked okay for many years..

Today there is a bad, constant hum on all phones. When I plugged a
phone into the outside box jack, the hum was the same. It has also been
raining for several days.

Can a bad ground cause a hum? Vigorously shaking the ground wire caused
no static. What should I expect if I disconnected the ground entirely?
(As I write this, I'm thinking maybe with corrosion the path to ground
is now non-existent.)

Also, there was a wasp nest in the customer side of the outside box
(but not in the company side). The nest didn't seem to create any
obvious destruction, though. If I connect a phone to the jack on the
customer side, then the nest seems to have no involvement anyway since
the terminals on that side become irrelevant - but you never know.