Yes, you can! But what is given away is carved up into very tiny pieces, called Satoshi... 20,000 of them are worth about .20 at a btc price of $1000, if i've not messed up the math... so when you get 20 or so Satoshi, it takes a LONG time to get anything.... and usually they have minimum payouts of 5500 or more.....
Check out Land of Bitcoin Faucet.... it will walk you thru about 50 different sites giving away satoshi....... makes it easier...... most of the sites there pay to one of two "clearing houses", which hold you satoshi until you have the minimum amount (coinbox.me and microwallet.org). and then they send the payments to your bitcoin wallet..... I have had NO difficulty with this in about 2 weeks now...... it is a huge investment of time for a few pennies, however... beware.
But it's kind of fun, and some of the sites are interesting. Do it for entertainment, not a living!
I've not run across anything or anyone paying out a substantial amount of bitcoin. If you find it, please remember where you got this, lol