Actually, you can upgrade award tickets to the next class and pay with miles for the difference in fare IF there are available awards.
I just did this with our return trip from Europe coming up this summer. I redeemed SAAver Econ tickets one way for our family, miles came out of DH's account (3 x 30K AAmiles). Then I stalked the AA website daily for Business SAAver tickets with the same origin AND destination (you can change AA award tickets dates & times with no fee as long as the origin and destination remain the same). One day I was lucky and found Business SAAver tickets (3 x 57,500 AAmiles) available and called Customer Service to make the change. I was able to upgrade us to Business class (difference of 3 x 27,500 AAMiles), but was short a few thousand miles and had to buy them then and there over the phone. AA was running one of its bonus mile sales, and I bought the lowest amount that earned bonus miles, paid for it on the Aviator card (which incidently earned 2x miles on the purchase). The taxes and fees were about $15 more on this itinerary so that got paid too - the original amount paid got credited back to the Citi card used originally and then the new taxes and fees were charged to that Citi card. It did take about a week for all the changes to be official, and a phone call to Citi to approve the new charges (I don't know why they flagged them, I mean c'mon it was a Citi AAdvantage card used to make a purchase at AA).
I would think you could use this for AAnytime awards too, since those are always available but also very pricey awards tickets, so you'd need to see if the math makes sense.