I've been trying for several days to buy train tickets for our trip to Spain. I keep getting this error:
En estos momentos se está realizando otra proceso de compra. Por favor, espere a que dicho proceso finalice antes de iniciar uno nuevo. (CE07)

Which google says it means: At the moment another purchase process is being carried out. Please wait for said process to finish before starting a new one. (CE07)

Any tips on booking train tickets on Renfe? I am using an US-based credit card. Thank you.

Use Paypal.

Okay, I'll try that, thanks.

. . . or sometimes it just a matter of opening a different browser, or clearing your cache.

jansj, I've cleared the cache several times, no success. PayPal worked, although it's not my preferred method of payment.

Does your browser block pop-ups?

If yes, try a browser which does not block pop-ups. Maybe during the verification process it pops up a smaller window to ask a few security questions, and you'd need to answer those questiions in order for the payment process to continue.

Or try thetrainline.com ?

Reading54,
No, no pop-ups, but from what I've read it has to do with US credit cards. For some reason, the Renfe website does not accept them. PayPal worked, and I'm not sure if the conversion to USD is done by Paypal (DCC) or by the bank.