I\'ve just been looking through a website\'s error_log and one of the error\'s that has been logged a few times is:
[21-Jun-2011 12:24:03] PHP Not
Yes, it's possible, but it never happens for a regular request.
All browsers do send a browser string in the request, so any request that arrives without one comes from some other program. Even all well-behaving bots send a browser string, so you don't have to be concerned about not showing up in search engines either.