Sometimes when I sneeze (and sometimes at other times, as described
below), I experience an almost debilitating pain in one or both arms. 
The pain is not a cramp, but feels as though it travels through the
entire arm, starting at the top and quickly moving down to the
fingers.  I can move my arm(s) when this happens, but I'm strongly
disinclined too.  It's a fairly severe pain that may last for 30
seconds or so.  Sometimes it's only one arm, and sometimes it's both.

7 years ago, when I was 20, I spent about 6 months eating a total junk
food diet and being very unhealthy.  During that time, I would often
get these arm pains upon little provocation - for instance, while
reaching to pick something up.

Since my diet has improved, I generally only get these pains when I
sneeze (and not every time I sneeze), or occasionally if I cough.

I recognize this advice cannot take the place of a diagnosis from a
doctor, but, any idea what this might be?  Is it common?

Hello,

I have researched your question and have compiled some information for
you. I was unable to find anything that suggests that diet might be
associated with the pain. The improvement might have come from a
different change in your lifestyle (excercise) or it could have been
your diet. I was able to locate several people who have your problem
(pain in arms or legs while sneezing or coughing). Please take the
research below as information only and see your Doctor for
professional advice.

The best information that I was able to find for you was a reply in
the Neurology and Neurosurgery Forum on http://www.medhelp.org
"The second common cause of importance is cervical radiculopathy, or a
pinched nerve in the neck or cervical spine. In this situation the
pain is often related to neck movement, or to coughing, sneezing, or
straining."
The post recommends that "You should see your primary MD, specially if
your pain is persistent or getting worse."
Source: http://www.medhelp.org/forums/neuro/archive/8008.html

The response outlines three causes of pain in the arms, the second
which appears to be the best possibility.

Have you had any neck or back injuries? If so, you might be interested
in reading the following:
"Weakness or trauma may allow the inner portion to push through the
outer ring (in some cases compressing one of the spinal nerves and
causing pain and numbness in the neck or along an arm or leg)."
"Herniated Intervertebral Disk"
http://www.hopkinsafter50.com/html/silos/backpain/libHerniated.html

It appears that this is usually accompanied by back or neck pain and
is more often found in people over the age of 45, but it could still
be a possibility.

The following website suggests that chiropractic care might be
beneficial to you:
"The symptoms then, are all the more recognizable and made more severe
by coughing or sneezing. Nature responds to the incessant pain by
varying degrees of immobility. Shoulder and arm pain constitutes the
most frequently recognized symptoms of the lower neck and upper
spine."
Source: Neck, Shoulder, Arm Pain 
http://www.woodburyct.com/woodburychiro/neckshoulder.htm

I found a person who seems to have a similar symptoms to yours at 
"Pain in Arm While Sneezing"
http://neuro-www.mgh.harvard.edu/forum_2/PainF/12.4.996.53AMPAININARMWHI.html

There were two replies to this post which can be found here:
http://neuro-www.mgh.harvard.edu/forum_2/PainF/12.7.99.html
http://neuro-www.mgh.harvard.edu/forum_2/PainF/ACompressedNerveOrSpinal.html

Search Strategy:

pain sneezing arm
://www.google.ca/search?hl=en&ie=UTF-8&oe=UTF-8&q=pain+sneezing+arm&meta=

"pain while sneezing" arm OR arms
://www.google.ca/search?hl=en&ie=UTF-8&oe=UTF-8&q=%22pain+while+sneezing%22+arm+OR+arms&btnG=Google+Search&meta=

sneezing coughing "pain in arm" OR "pain in arms"
://www.google.ca/search?hl=en&ie=UTF-8&oe=UTF-8&q=sneezing+coughing+%22pain+in+arm%22+OR+%22pain+in+arms%22&meta=

Thanks for the prompt answer :-)

the symptoms highly suggest pressure on nerves leaving the spinal cord
in the neck: at your age, most likely a bulging disc in the vertebrae.
Chiropractic manipulation without a proper diagnosis could be
dangerous. Likewise, it's important to find the exact cause of the
pressure; it may be, when fully understood, simple measure may
suffice. But it sounds like something that needs an exact diagnosis;
which means seeing your doctor, and likely getting a MRI

I am 22 years old, eat a fairly well balanced diet, lots of carbs, no carbonated
beverages, lots of veggies on my whopper.  I am could lose 30 pounds,
but I get around just fine; I load trucks for ups- a fairly physical
job.
  I sneeze wrong.  It hurts when I do, I can almost watch the pain
float down my arm into the palm of my hands.  I used to believe that
the pain was caused by capillaries expanding as a non-escaping sneeze
pressurized by artieries.
But the nerve theory would make sence too, because sometimes my hands
will hurt at work.  But I still say it feels like they need to be
pressurized and heated.
But that's just my interpretation of the signals the body puts out.
who knows, maybe ups will foot the bill for the neck thing.
-Seankubin