On a lot of factors.
First, there's the game style. If you're making Angry Birds 2 then, yeah, it would probably be in your interests to make sure it runs on everything. Your target audience may have a rather crummy computer and the larger you can cast the net, the better. Conversely, if you're making Quake 6: Gorestone Bloodbath, then you can probably get away with not running on a five-year-old platform. Your target audience wouldn't be caught dead with anything that old.
Second, there's the sheer game requirements. If you're making Angry Birds 3 and your design doc requires that each individual plank of wood be able to split apart into properly-simulated splinters, then your users will have to have a good computer, and therefore you can probably rely on your users have a reasonably good graphics card. (Especially with the new Intel integrated graphics.) Conversely, if you're making Bejeweled 3 . . . well, you could run Bejeweled on a wristwatch, so you don't get a free pass to say "yeah, we'll just require more".
Finally, there's your own resources. A one-man game studio doesn't have much time to spend to make sure the game functions on absolutely everything under the sun - it may be perfectly sensible to say "okay, we're going to require opengl 2.0, we'll pop up an error message that recommends updating drivers, we're just going with that." Conversely, if you're a 200-person AAA studio, you may find yourself doing compatibility tests on specific individual network cards.
In the case of Angry Birds, I would personally implement a fallback to split high-res textures apart, at the cost of some efficiency, and go with that. If I found that a significant number of users were hitting the fallback and then having a bad game experience - but not such a bad game experience that I felt it was unsolvable - then I might spend time on a better fallback. But I'd be spending far more time making a good game than trying to eke out percentage points of potential market share. A great game that 95% of humanity can play will sell a hell of a lot more than a mediocre game that 99% of humanity can play.
And finally, if you're an indie developer making artistic games, remember that the long tail is your friend. If your game is great, ten years from now people will still be buying it, and your "OpenGL 2.0, 2gb of RAM required" message will seem laughable compared to the bargain-basement 256gb RAM OpenGL 6.0 machines that Wal-Mart will be selling. I don't think this applies as much to the AAA or casual markets, though.