Obviously this quandary doesn't apply to some stats. Getting a strength of 24 isn't difficult. You just take on a Half-Ogre, a Background which increases his strength, and by george you'll probably get a Strength of 21 in very short order.
There's some stats that just aren't catered to by racial modifiers, though, and as I understand it those modifiers can be pretty crucial. Intelligence is my primary concern, but Perception is likewise a curious little monster to raise. (And the bonus for that one's theoretically pretty awesome, if terribly niche.)

Intelligence. My next character is to be a technologist, thus requiring an Intelligence of 19. While there are means to temporarily increase one's Intelligence, if I sought to increase his Intelligence beyond 20 it should be only one point and one bonus out of my way. Where, though, to find this extra point? What does work, and what doesn't?
Perception. While this isn't too crucial to any character build I have in mind, nor did my last character have any particular difficulties with invisible attackers, nevertheless I feel a little miffed that I cannot determine how to raise it past 20.

So what does and doesn't work for this?

Okay, two things come to mind. First of all, it seems you don't need a 21 for these to be effective I misread the manual. (Pg. 18, Arcanum Player's Guide.) It says these are the bonuses for meeting or exceeding 20, not for exceeding only. Therefore, if I want to pump one more point into Intelligence after mastering a field of technology, that one additional point is enough for the +10%. So you were right about that.
That said...
To clarify, you're quite certain that backgrounds add to one's maximums? 20 is, after all, a nominal maximum for a stat. Have you tested this? At this point, I'm mostly just curious if it's possible.

Well I didn't test them all obviously, but for one of my characters I took the "spoiled childhood" background that gives +6 to willpower and I could take willpower beyond 20.

In that case, I'm definitely going to have to give this a shot. This calls for a divergent save game!
For now, though, thanks.