Absolutely possible...in fact, I have that exact situation with a soldier I 'follow'...
Mine was a Pa. soldier as well....He enlisted in the 7th reserves at the beginning of the war, served his 3 yr term...was discharged...and then re-enlisted as a substitute about 8 months later...thus, he was in a different / new unit. It also cost him his rank. He was discharged form the 7th as a Corporal...he re-enlisted in the 100th (the Roundheads) as a private.
All of these circumstances wreaked havoc on his later pension fight. When I physically pulled his service and pension files down at the Archives, I was delighted to find a stack several inches thick...Bad news for him ( a heck of a bureaucratic fight), great news for me as the amateur historian / researcher...
In post #4 of this thread, 'mcfb50' is spot on....
That card you posted is from the state (Pa.) archives....You have to realize that at the beginning of the war the Federal record keeping was nonexistent to p*** poor, at best.....So, when the govt. began to look for these guys service records...they would find nothing to corroborate their mustering in....and would send requests to the various states for their records (if any)...I have paperwork specifically stating that this happened in my case study's situation ("applicant's federal file does not contain much [read: muster rolls merely stating 'present']...request sent to state archives for soldier's service and muster records)....and in my case...the state did not have much either...So, he had to gather 'sworn affidavits' to support all claims...including documentation of his birth...which was supplied by a cousin who had a family bible with a hand written notation of his birth....and then get an affidavit as to the veracity of said cousin etc, etc, etc....and on, and on, and on it went.....Lord......