I warn people to proceed with great caution when thinking about buying an airplane as a student pilot, but along with my warning I like to work out facts and yes it is "possible" you might save money (it's also possible you might pay a lot more)... the main reason to buy an airplane is because of the freedom of having an airplane available to you whenever you want it and the simple pride of owning a flying machine. There are definitely some good deals available, but there are even more neglected airplanes sitting around that haven't been flown regularly or maintained well and the owner is trying to dump them cheap to get rid of the headache.
I currently have two student pilots I'm training who own their own airplanes, one actually bought a plane before he'd ever even ridden in a small airplane!
As a student you likely know nothing about airplanes and could end up buying something that needs a lot of work, or you might end up buying a very sound airplane. You should have a pre-purchase inspection completed by a qualified A&P/IA and have them complete the inspection as an Annual Inspection if you decide to complete the purchase. Then you know the status of the airplane and can fly it with confidence for the next year.
With a Cessna 150 (and many other tricycle gear small planes) expect to pay somewhere around $750 per year as a student pilot for quality insurance. Estimate $1,000 for an Annual Inspection with a few small repairs (if there are problems this can go up quickly). Tie down or hangar will vary a lot by location, but if you assume $500/year for cheap storage (tie-down) you're now at $2,300 per year before it's moved.
At ~5.5 gal/hour you're paying $35/hour for 100LL or $18/hour for ethanol free autofuel (after paying $150 for the AutoFuel STC if it doesn't have one already). Add $2/hour for oil changes (doing them yourself).
On a simple airplane with relatively low time on a quality engine you'll want to add in a $15/hour engine and maintenance reserve because at 1,500-2,500 hours you'll need to get the engine overhauled and somewhere before then you'll probably get one or two top overhauls and cylinder replacements, and various accessories repaired/overhauled.
So, if you fly 200 hours per year (4 hours per week) and could find a 150 for $80/hour wet you'd pay $16,000 for rental with no maintenance worries to deal with.
If it's your airplane burning auto fuel you'd come out to $35/hour for fuel, oil, maintenance and another $12/hour to cover fixed costs, for a total of about $9,500 for 200 hours. A $6,500 savings. Now, you might break some stuff, but $6,500 goes a long ways on a Cessna 150.
Now, if you were flying 75 hours per year... the picture changes quite a bit and it starts to be about the same price... until something breaks and then your own plane is costing you more. Also, if you dont have a reliable and convenient way to use auto fuel and are paying $6+/gallon for 100LL... the cost savings are not as significant.
Also, dont forget sales tax and other state fees which could add $1,000+ to your acquisition costs for a cheap plane.
Edit: Note also I'm assuming a cash purchase, where you're looking at this as an equitable money pit. If you're financing it and making payments.... ick.