My mom owns a 15 year old Toshiba CT-9534 29inch colour television.
For the last few weeks, instead of seeing a whole image on the screen,
the bottom seems to be compressed and cut off, and finishes at the top.
I'm not sure how to fix it, although I think that after a decade and a
half of service she should retire her trsuty steed for a new one. Any
ideas?

Probably bad electrolytic capacitors in the vertical circuit.

If the color and brightness still looks good, then the tv is probably
worth having repaired. Typically with those symptoms one or more
capacitors have gone weak in the vertical deflection. A very basic
fault that is easy (relative to skill) to repair and usually not too
expensive if the tv gets carried into the service shop.

In home repair will add the trip charges which are not inexpensive.

If she were to replace it, then she would have to get use to working a
new tv set. Make sure she is completely comfortable with all the
features and operation of any replacement tv she might want to buy
before it gets home. Do not simply throw that tv away, it was a well
built unit and is worth having repaired by someone to keep it out of
the waste stream a few more years.