Well, I have had good luck sometimes and bad luck sometimes when trying to buy tickets directly on the RENFE site. Based on what the "man in seat 61" says, I think the reason may be that RENFE requires "verified by visa" for Visa cards, and I have just learned that you can only activate that feature on Visa debit, and not credit cards.
Sure enough, last year when I purchased my tickets on the RENFE site, I used a Visa debit card and had no problems. This year, I tried to use a Visa credit card and it wouldn't take it. So I went over to raileurope.
The only advantage to using the RENFE site directly rather than raileurope.com or petrabax.com (tip from alipilgrim) is that you save the service fee, which is around $7. But as of this year, the cheap RENFE web fares are available through raileurope and petrabax, which means it is very much worth the effort to make sure you buy the ticket exactly 60 days ahead of time. The cheap web fares sell out very quickly.
Two additional tips -- if you're at the very end of the month, say on the 29th or 30th, it's worth checking to see if they've opened the seats up for 61 or 62 days in advance. And, if you're in the US, remember that it will be midnight in Spain some time in early evening here, so you can actually buy tickets on the 5th of April at 6 pm central time for the 6th of June, because that's midnight in Spain.
Buen camino, Laurie