You can extend your visitors visa (which is not a "VISA" per se, just the right to stay 3-months with entry on your passport) from 3-months to 6-months but that's it. You'll have to go through a formal "process" to for this so it's NOT automatic.
If you want to stay longer than the total 6-months you'll still have to return to the USA before being legally able to stay in Spain because you have to ENTER Spain with the Visa in-hand. You can't simply stay while the visa is processed - whatever visa you might be getting (student, work, etcetera).
As madridmadridmadrid above mentions, getting a student visa would be the easiest way to study but you'll have to pay for your classes. And while many are inexpensive, we're still talking about several thousand Euros total. If your fiance is studying for her masters NOW I'd recommend you start your own student visa process as soon as possible in order to be able to return as soon as possible. Is it all a big pain in the neck? YES! But that's the way it is. Sorry! And good luck.
Saludos, MadridMan
_________________________
Visit BarcelonaMan.com for Barcelona information, Transportation, Lodging, & much MUCH more!
Curious about what could POSSIBLY be inside the brain of MadridMan? Visit
MadridMan's Madrid Blog