As others have noted, grey telephone box (NID) contains a surge
protector that is effective if earthing connection exists. Phone will
work with or without earthing wire. But without that wire, then human
and transistors are both at risk and phone installation violates code.
Telephone installations must have that 'whole house' type protector and
earthing. Earthing wire must be continuous, no splices, not bundled
with other wires, short (ie 'less than 10 feet'), and connected to same
earthing electrode used by AC electric and cable company ground wire.
As others noted, you want that earthing wire corrected - and without a
splice.

Hum may be from a partially damaged surge protector inside NID.
Determine whether hum is inside house or in NID box. By doing as Alan
has recommended, open 'Customer access' side where your wire connects
to their box via RJ11 phone plug. Simply disconnect your wire and plug
a telephone directly into that phone jack. If hum is heard, then
problem is on telco and they pay. Otherwise problem is on your wires
and you pay (or fix it yourself).

That 10 AWG earthing wire (they are responsible for it) must be
fixed. But first determine where problem exists - on your side or
theirs.

wrote:
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.)
...