Is it possible to get addicted to sleep?

Not sure: I'm not sure what you mean by addicted. Everyone needs sleep and if one doers not get enough sleep it will affect that persons ability to function and stay healthy.

Sleeping problem: I don't think it is addiction to sleep. Everybody needs to sleep otherwise it will not only affect the function but also can cause other problem- including obesity etc. Now, is somebody sleeps too much- i don;t think it is an addiction. There should be a reason why that happens- whether it is due to sleep apnea, other sleep problem-narcolepsy etc, anemia, depression etc. Go to see md and d/w md.