I can vouch first hand you can do it, as well as the parents sip provider mentioned is a fine choice as I also use callcentric, and its a fair price imho. you can even call port your number to them (ATT however..... will Drag... Their....Feet... then call port you on a Friday after business hours).....
I have a bit different setup from the parent--mine utilizes a PAYG number when i have no wifi access. If i am not logged into callcentric, they I can make a rule to auto-forward to my PAYG number.
Get a sip capable pone, honestly, pay the extra bucks to get a smartphone, versus standarlone sip phone with no features. you'll like the web browsing and email when you have wifi. (Note: most nokia wifi capable UNLOCKED phones have SIP integrated... I use an e71)
Voice works wonderfully for me, but people get a bit confused if i call them back from my PAYG number, and if you really need to text, callcentric doesn't support sms. but you could from the PAYG number, or from a free Gvoice account (additional confusion)
in total, my bill went from ~54 per month plan on the cheapest 450 minute plan with sparrse texting, to somewhere around ~11-14 per month. I have paid for the smart-phone two times over at this point.
I'm pretty techy, and I looked into a PBX solution as some have mentioned, but either you're stuck with a non-static number, a non-call ported number (google) number, or to do it yourself your rates are fractions of a cent within callcentrics cost to you. combined with the power of said device and annoyance to set it up, and be your own telco, I decided to not bother with the self-pbx route. If you need some very fancy call wizardry, then maybe a in-house pbx would make more sense. (such as screen everyone but a group of numbers after X time is what i would like)
one more thing, do you talk a lot? cellular may still be better for you. I estimate my usage around 300-800 minutes a month usage.