My watch isn't running right. Someone said it might be 'magnetized', is
this true?

Possibly.  Magnetism has a more adverse affect on older
mechanical watches.  Wrist and pocket watches made before the
1930's (give or take) generally have steel hairsprings with split
bi-metallic (brass/steel) balance wheels.  When this type of watch
becomes magnetised it will run much faster because the steel
coils of the hairspring will stick together.  Watches made ofter
this time had hairsprings and balance wheels made out of highly
non-magnetic alloys (Invar, Glycudur, etc) which are resistant
to magnetism.  

Now I'm sure many of you remember your father's (or
Grandfather's) watch they had and you never heard much about
their watch ever being 'magnetized' when they wore it.  Now
suddenly, now that you have it, it may have become
'magnetised' (sometimes more than once), how?  It's quite simple
when you think about it,
magnets!  Or more importantly magnetic
generated by things like cell phone, microwaves, computers
& computer monitors, TV's, radios, speakers, or basically,
anything that requires a
magnet.  In you parent's (or
grandparent's) they probably kept their watch on the dresser,
maybe in a jewelry box, not on top of your bedside clock/radio
(magnets in the clock & speakers), we use cell phones &
microwaves and other things as part of our everyday lives and
we don't give a second thought as to how this may affect our
Take heart!  Watches can safely be demagnetised without any  
lasting problems.
 Now, newer watches made with modern
non-magentic alloys can become very slightly magnetised.  Even
though they are billed as 'non' magnetic, it is still possible under
certain circumstances to magnetise them, even though only
Again, this is not a permanent condition.  

Quartz watches, of course, run on batteries, which means that
the battery runs an electric motor.  Electric motors create a
magnetic field, the motor of this watch is a permanent magnet.  
Therefore, if you were to de-magnetise a quartz watch it would
stop working.