Actually, it is pretty common, especially among non-trauma doctors like Paul (he was just an OBGYN). My dad was an electrical engineer. He joined the Air Force to avoid being drafted into the Army, and guess what, he was loaned out to the Army in the field as a radio operator. The point is, Paul lied both about his service and about Newt. (and it isn't the first time, he isn't the nice old honest grandpa type his supporters make him out to be.)
No it is not common at all. I have forgotten the indecent but in WW2 or WW1 a great scientist was drafted and killed in action as a regular infantry man. After that either a law was passed or an executive order released that stopped the practice of drafting certain categories of occupations into a a regular infantry MOS. Medical Doctors being one of them.