I don't know why they do this. I think there just comes a time in a cats life, where they just don't care anymore.
I'm dealing with a similar thing right now with my 12 year old cat, who has never used a litter box. I'm at my wits end with it though, I can't handle it anymore. I've tried different litters and she doesn't like any of them. So I got rid of the litter box and put them outside during the day. It was fine during the summer and two of them still go outside all the time to potty then come back in to sleep. The one that goes in the floor, makes no effort to go out at all anymore. I think she is senile. Sorry, she just peed in my floor and I'm fuming right now!
Anyways, back to YOUR problem! I think maybe you should change all the litter maybe more than once a week. Scoop everyday still, but do a massive change every like 3 days or whatever, like Wednesday and Sunday. What kind of litter are you using? Sometimes they just don't like the way it feels on their feet and sometimes I think they just get sick of it. Does the box have a cover on it? Maybe take the cover off, or if you use any litter fresheners, stop using them. I only have one cat who constantly used the litter box, that was nice. lol
Sometimes cats drive me up a freaking wall!