I noticed mine do the same thing. Mine actually died a week or two after the breaking up and turning light green. I think it was the lack of nitrates or food that caused it. When I added it to my sump my trates were 30-40 ppm. My water now reads 2ppm for trates. It slowly died as my levels went lower.
My nitrates are pretty high 30+ right now. Ive only had the refugium setup for about a month and just got my cheato 2 weeks ago via ebay user. Its not really all falling apart just brittle and breaks when handled into small pieces. Its probably fine, I was just wondering if all cheato is brittle like this? Its a pretty dark green color. Like dark green not light green. I will keep an eye on it and just NFWI.