Breaks from meat seems natural. Unless a caveperson always succeeded in his/her hunt (or was 100% accurate with food storage techniques, or never experienced draught, or was never injured, etc).
Part of the reason fasting might be good is to help protein recycling. Are there other reasons why it might be good to take a day off of eating meat every week or two? I'm thinking that there might be reasons that science doesn't know about yet, since we barely even know what optimal levels of vitamins and minerals are. Maybe something involving hormesis?
Eating meat every single day is a great way to get into paleo, but I don't know if it's what our bodies are 1) used to, or 2) optimized for.