Maybe - definitely enough power for a HD 3850. Of course, depending on what else the PSU is also running, you may need to take out a couple hard drives, case fans and the CPU to power the video card.
On the other hand chances are you have an "average" system and could be OK with a 450W PSU that provides 216Watts (18Amps x 12 volts) for ALL the 12v parts in your PC. To be honest even a lot of the cheap 450W PSUs output 300+Watts of 12v power.
If your PSU is a couple years old you'll need to be careful about what else you put in your case as well as being careful about OC'ing your system.
Now wouldn't it have been easier to put more detail about your PSU and your system in your post?
Everything I have seen is that the 3850 uses ~100 watts. Under load, a 3850 in an average system would use 250 watts total.
For what its worth, I have a Shuttle case with 350 watt PSU, and I will be going with a 3870 shortly. People that I trust tell me that I should be more than fine. (and the power use calculator link confirmed this)
This is what I found at Newegg under several companies specifications for the 3850:
PCI Express based PC is required with one X16 lane graphics slot available on the motherboard
450 Watt or greater power supply with 75 Watt 6-pin PCI Express power connector recommended (550 Watt and two 6-pin connectors for ATI CrossFireX)
Certified power supplies are recommended.
Though I'm waiting for income tax time to get my two 3850's (one for each PC), I did upgrade the power supplies to a 550 watt Antec True Power in my wife's PC and an Antec Neo 550 watt in mine. The Neo has 3 12 volt rails with 17 amps each and the True Power has 3 12 volt rails with 18 amps each.
For all practical purposes, the rails act as one when supplying power to the graphics card. A single rail power supply would list more amps on the rail than are listed on one out of 2 or 3 on a divided rail power supply. There's thread here at Tom's that provides a formula to calculate the amps from maximum amps given by manufacturer's but I don't have it bookmarked.
Generally, make sure that your power supply has at least 34 amps maximum on two rails (17 amps on each). Since a 3850 uses less power than an X2900, many 450 watt name brand power supplies would work with one card today, as I've seen 450 watt power supplies with 29 amps, but older 450 watt power supplies might not provide enough power at load.
I moved a 450 watt Coolermaster Realpower over to my 7 year old's PC because it only had 22 amps on the 12 volt rail, and I wouldn't trust a 3850 on that, not under load. Never trust an off brand power supply that comes with a barebones. The old fashioned test is the weight of the power supply. Good internal components weigh more, I've tested that by hefting a 450 watt psu from a barebones I bought in one hand with the 450 watt Coolermaster I replaced it with.
It's better to have a power supply that can handle a bit more than you need, because a barely capable psu might run under load all the time, whereas a good name brand psu runs stable. Also always buy a psu with active pfc. Many new psu's are 80% certified and that's good.