I don't think my $10 computer speakers are up to the task of producing test tones reliably at any frequency... I think a lot of people are testing their speaker abilities more than their hearing with this test... or with any online test. If I go through and match these tones to the same db level on each freq to my ear, I end up looking at the typical curve of crappy speakers... Rolled off highs, boost in upper mids, big hole in mid bass, etc.
As to headphones, I don't know.. but you're plugging them into a 2 cent connector on a noisy motherboard.. so I don't know if you'll be able to discern computer whirring and general noise from 10,k up. Most general audio systems are capable of producing 1 to 6,kHz fairly well.. and it sounds to me it varies 8 db in that short range. (generally) This is generic stuff, the 5 to 50 dollar headphones, systems, and so on. I would imagine a cheap boombox is as good as most PC speaker systems.
If you're listening to this type of test in an isolated room with headphones on, with a completely separate audio system that's dead quiet on really nice headphones, you might get a better idea of your hearing limits.. but even then I'm not sure I trust the source of anything online for this purpose. A test tone generator directly to headphones is best.
BTW, pretty much every ENT offers free hearing exams. Let them know you want to be tested to 20,k.
-C