I was a short-termer in the Navy and took the middies on shakedown.
I received an Honorable Discharge from the Regular Navy and signed up for a six or nine-year period for Inactive Reserve.(Can't remember which.) Received another HD from the Reserve before that period expired when I joined the Air Force in the mid-50's.
My only point here being but to state that for enlisted personnel, real paper Honorable Discharges are given when a discrete service period has ended, and that you can't be a member of more than one branch of service at a time, (can't draw double pay) though aggregate time accrued can count towards retirement -- even reserve time. Is that correct?
Thank you for your kind words. Things are a helluva lot different now. Paint fumes, bad. "Human Capital Index Score", good. (Or so I'm told....)
Anyway, you're correct on discharge certificates. All us enlisted pukes get a nice little geedunk certificate at the close out of every hitch dated the day before we reenlist. Officers get one at the end of their officer candidate/service academy time, and, eventually, when they resign their commissions. Those of us who've endured the slings and arrows of outrageous fortune for 20 qualifying years of service are transferred to the Fleet Reserve until we reach age 60 when we are ultimately retired. The intervening years we are on retainer... and aggravating the hell out of docs at the clinic, the young ones in the exchange, and our wives (who generally aren't prepared for us to be on permanent shore duty).
And, no, one can't serve in more than one branch of service at a time. A good Personnelman will get into a fine lather if the dates of service for servicemember who's come from another service or the reserves are just so. E.g., "Shipmate, you just CAN'T have enlisted in the Navy on 01 Oct 2003 when you weren't discharged from the Air Force until 01 Oct 2003!" No gaps! No overlaps! Makes the pay guys even more apoplectic.
And, yes, all your time counts for pay and retirement. Except for delayed entry time (the waiting time from enlistment to boot camp) of most. Some of us got a few months credit for it. However, the active and reserve retirement systems are different and THAT is course of instruction all unto itself....