On both of my radiators that fit underneath bay windows, the bottom half
of
the radiator in the middle (long) section does not heat up. Both ends of
the
radiator, and the top of the middle section does get hot.

That is usually sludge in the radiators. Not only should each radiators be
totally cleansed (either by taking outside and hosed, or using the often
overpriced powerflush), but afterwards ensure that you run for a week or two
with a desludging chemical. After two weeks, flush out following the
manufacturer's instructions and refill with a corrosion inhibitor.

It was this lack of inhibitor that probably led to your system's current
sorry state.

You may wish to find out why there was insufficient inhibitor in your system
so the problem doesn't recur. Some possibilities:

1. The system has a leak, so the water (and inhibitor) leak away and is
replaced by fresh water (sans inhibitor).
2. The system occassionally or constantly ejects water through the
overflow/pressure relief.
3. A misguided plumber couldn't be bothered to install some, perhaps in the
hope of drumming up business, or, perhaps, through complete ignorance.
4. The inhibitor is so old that it is no longer effective.
5. An insufficient quantity, or inappropriate type of inhibitor was used for
the type of system.

Christian.