There is a limit to the number of roms you can add using the Edge card, and then it stops displaying any new ones in the menu screen. Total guess and probably not right but could there also be a limit to the number of saves you can have on the SD card? Like I said that's probably total crap.
Anyway, does it do it every time you save? If you delete the .sav file and then try saving a new game does it work and then not work the next time you try to save or is it every time you do it? It's not something really simple like the SD card is full without you realizing?
What you could try is see if you have another SD card, maybe in your phone or somewhere, and back it up and try that, just to see if it will work on that. If it does it's probably a problem with your current SD card. I have to say when I had problems with an old SD card the games just froze up, they didn't carry on moving in the background, so it's a bit of a weird problem.
Sorry if these ideas are a bit obvious.