Valda, note these three different verb pairs:

1. просыпать/проспать -- to oversleep
2. просыпаться/проснуться -- to wake up
3. просыпать/просыпать -- to spill something dry and granular (salt, sand, seeds, ashes, etc.)

With the help of Google and ru.wiktionary.org, I found some information about their etymologies:

The first two verb pairs, relating to sleep, are Indo-European cognates with Latin words such as sopor ("sleep") and Somnus ("the god of sleep").

The third one ("to sprinkle dry stuff"), is cognate with the -sip- in "to dissipate", which ultimately comes from the Latin verb supare, meaning бросать ("to throw").

So, in short, you've got the bases -спать, -сыпать, -снуть that all connect with the noun сон ("sleep").

And then you've got a totally separate base -сыпать, which refers to "dry sprinkling."

P.S. I was a bit surprised to find out that the English word "sleep" is NOT cognate with the Russian спать or Latin sopor ! Instead, it apparently comes from an IE root that signified something closer to дряблый ("limp, wilted, flaccid").