7:46 p.m. on December 29, 2007 (EST)
Just for an amusing counterpoint (real, but don't take this too literally), the tent I was supplied for my Kilimanjaro hike last week (camps at 9900, 12672, 13035, and 15180 ft elevation) was a 2-person 3-season tent (guyed, non-"free-standing", lots of mesh for the inner tent, but fly down close to the ground). I used my -40 deg down bag (microfiber outer cover). It did snow (and rain and sleet), though the low temperature in the camps was probably never much below freezing (the inside of the fly got LOTS of condensation, and at the highest camp was heavily coated with frost). The summit of Kili is 19475 ft. As one of the infamous "7 summits", this is often considered a full-on expedition. Yet the tent was, frankly, a pretty low tech 3-season tent. The brand name is unknown in the US or in Europe (it is sold only in Africa).
I have also used a Sierra Designs Flashlight in full-on winter conditions in the Sierra (subzero, blizzard that dropped 2 feet of snow). Again, non-"freestanding", lots of mesh, and yes, the snow drifted in through the mesh.
These are two cases where many of the above comments have been pretty strong that "expedition" tents are "required". But do not misinterpret what I am saying. The fact is that you do not "need" a full-on expedition tent (I would normally use one in the conditions the original poster specified). You CAN "get away with" a "sturdy 3-season tent" in North Tejas conditions, though it can get pretty brutal there (been there, done that, don't ever wanna do it again! - talk about your winds roaring down over the Plains, kicking up multi-thousand foot high dust clouds and dumping way too much snow on a supposed desert!). I don't know where you want to camp in the Land of Enchantment (your neighbor state), but when you get into the Sangres, you can get deep snows and real blizzards (Taos has some of the finest deep powder skiing in the world, but the Sangres can get some tough blizzards, too).
The main thing is, if you watch the weather forecasts and pay close attention to the changes while you are out there, you can be just fine in your "sturdy 3-season tent". But if you push it to the limits of your personal experience (note I said "experience" , not limits of your gear), you risk getting in over your head (literally). You are talking about learning "winter" in an area that can have real extremes, but one in which with judgment, you can proceed by easy steps and learn your personal and your gear's limitations.
By the way, calamity, one may "hear" of accumulations of 6 feet of snow in a day, but the published climate records show these are pretty rare. The North American record for 24 hours is 76 inches in Silver Lake, Colorado, in April 14-15, 1921, with second going to Echo Lake, Calif (where I camp several times a winter) 67 inches, Jan 4-5, 1982. The single storm record for the US was at Mt Shasta, 189 inches, Feb 13-19, 1952. These extremes are rare enough, and predictable with modern weather data and tools that it is counterproductive to try to scare a prospective winter camper by quoting the all-time records as the expected "norm". By paying a little attention, the newby is far more likely to find a little "skiff" of a few inches, enough to make for a beautiful coating, but not enough to constitute a problem.