Challenge rules only allow you to recoup half your purchase price into the budget. But if we're talking about real life, it is possible to break even or turn a profit.
Last fall I stole... I mean, bought a 94 Saturn SW2 5-speed for $400. I put in rear struts, rear brakes, and drove it through the winter, keeping the Miata off the road. Cheaper than a hardtop! This spring, when I got my Sentra SE-R, I sold the SW2 for $800. I pretty much broke even on the Saturn, and the sale went a long way toward covering the purchase and rehab of the SE-R, which I find much more enjoyable.
Even longer ago, I was given - free - my first motorcycle, an 82 Suzuki GS650L. It was my friend's bike. Sat in his yard, unused, for 3 years, and then he was starting a family, etc. I put some money into getting it running again, and put my first several hundred miles of riding on it - without dumping it, amazingly. A small oil leak got bigger and bigger. Years later, I figured out that it wasn't engine oil, but gear oil where the driveshaft attached to the transmission, separate from the engine oil. I tore up so much in there that it was cheaper to buy a $500 GS550E I found. I swapped a few parts over, then parted out the rest of the bike on the GS Resources forum - turning a net profit!!! Even better, one of my buyers used a lot of my parts in a restoration that was eventually featured as the GS Of The Month on the site.
Once I was also given - free - a 91 Civic wagon, FWD, not RT4WD, but everything worked. I don't remember how much money I put into that car, but it was certainly less than the $1000 I sold it to a friend for. I felt bad turning such a profit off him for a while. Then a couple of years later he got $1300 for it as a trade-in. I've never felt guilty since.
So, yeah, a free car or turning a profit is possible, if you're lucky and play your cards right.