I know of nothing that explicitly forbids multiple backgrounds but the relevant sections of the Player's Handbook are written referring to a character's background, singular:
Every story has a beginning. You character's background reveals where
you came from, how you became an adventurer, and your place in the
world. Your fighter might have been a courageous knight or a grizzled
soldier. Your wizard could have been a sage or an artisan. Your
rogue might have gotten by as a guild thief or commanded audiences as
a court jester.
Choosing a background provides you with important story cues about
your character's identity. The most important thing to ask about your
background is what changed? ....What sets you apart from ordinary
people who share your background?
-- PHB p. 125 (Bold emphases mine)
Bear in mind also that backgrounds carry distinct mechanical advantages, and that these are balanced in a fairly straightforward way: A feature, two skills, and two language or tool proficiencies. (Also two personality traits, a bond, a flaw, and an ideal, but those are less important.)
-- Also PHB p. 125.
Giving someone a second background in the mechanical sense will be unbalancing.
Storytelling: With GM approval
If this is not a mechanical question but a storytelling/narrative question then, sure. The idea is that each character only has one background-- the character's own background. But there are two staples of serialized fiction that I have seen migrate into RPGs that run counter to that:
The retcon (or, more gently, the expanding background.) Writers, or players/GMs sometimes get cool ideas that lend themselves to an expanded view of a character's background. It happens.
The long, multi-faceted history, which really only makes sense by saying, "My character was and then he spent ten years doing and then kobolds ate his baby and now he's an adventurer!"
In my experience GMs often reserve the right to approve or disapprove that sort of thing, to keep players from going nuts. But if it has no mechanical advantage, then why not?
In Your Specific Case: No
Your specific case looks like one which is mechanical ("Would the character benefit...?") and is not truly a case of discovering a background, but just adding detail to his present situation.
So the answer is a resounding, "No." There is no mechanical advantage here, and there's no "background" here to speak of.