It is possible if you have the vast majority of the parts you need.  However, your choice of frame material and the fit you are looking for will be the biggest determinants of which route you should go, even if you do have many of your own parts.  If you know what kind of bike you need based on your fit setup, you can start looking for frames that will let you achieve that fit profile.  If not, you should get fit first so you know what to look for.

The most bang for your buck can be found by going with an aluminum frame.  They aren't the sexiest things out there, but you don't need that.  You need the bike to be functional and you need it to fit you.  Leader makes a pretty good $400 aluminum frame that looks pro and is pretty aero.  It also has a 78 degree seat tube which falls right around what most TT/Tri bikes will offer.  http://www.leaderbikestore.com/pd-ld-735tt-2009.cfm.  An even cheaper Leader option is the 720TT.  The frame isn't as aero and it doesn't look as pro, but it will afford you similar geometry for less than $150.  http://www.leaderbikestore.com/pd_ld-720tr.cfm.  Do a google search for some pictures of these bikes built up.

Fort Frames is another aluminum frame maker but the geometry is a more relaxed 75 degrees for the size medium.  This may not be the correct fit for you so make sure it meets your fit specs.  http://www.fortbicycles.com/html/komet.htm.

If you want some carbon fiber sexiness, your best bet will be to go with a well known brand that can offer lower prices due to volume and still provide a crash replacement program.  The Specialized Transition frameset will run you $1800 MSRP but you might be able to find a deal on a closeout from last year.  http://www.specialized.com/us/en/bc/SBCBkModel.jsp?spid=45893&eid=4338&menuItemId=12186.

For the same amount as the Specialized, you can buy an entire Trek Equinox 7, but it too has a more relaxed seat angle at 76 degrees.  http://www.trekbikes.com/us/en/bikes/triathlon/equinox/equinox7/

The Cervelo P2 frameset is listed at $2000 and is known as one of the most aero frames on the market.  http://www.cervelo.com/bikes.aspx?bike=P22010

If you are looking to stay under $1000, your best bet would be a frame similar to the Leader 735TT, but also to find an aerobar setup that isn't carbon fiber.  They will be cheaper and the weight you would save is marginal.  Your best choices for spending that kind of money will be in nice aero wheels and a solid aero frame.

So, depending on your fit requirements, personal preferences for looking pro, and the amount of money you're willing to spend, you can build a TT bike cheaper than buying a stock one, but you also need the know-how to put it together and have quite a few parts laying around in order to do it.

I have 5 bikes of all different types - mountain, road racing, touring, track, commuter/fixie. The track bike has wheels I built to match the bike. The fixie was an old racing bike that I completely rebuilt with a mix of used/new components. The touring bike isn't done yet but the components look like they belong together - similar styles. You can definitely make it look professional, the trick is to make sure the component styles match one another whether they're used or new and keep to a color scheme. Make sure that anything with color accents goes with the color scheme of the bike/frame. I usually pick 2 or 3 colors for the overall color scheme and then make sure everything with any color doesn't stray from those colors unless it's small and not too noticeable. The used components I give a thorough going over and cleaning to make sure they are clean and work like new.

Having said that, it's still cheaper to buy an entire bike instead of building your own, but then you'll be mostly stuck with what's on the bike. The exceptions may be the saddle and pedals. If you already have a good stock of parts it might be cheaper to build your own bike. To keep costs down try buying slightly used parts. You can get great deals if your careful (sometimes eBay prices are higher than retail) because a lot of people swap out parts that they don't want. To keep costs down I'd stay away from carbon fiber. I have aero bars that are almost just as light and were half the cost.

Good luck.