No, it will not. It is a common misconception that crt tubes go bad. The ONLY and I say ONLY company that has problems with the crt's in recent (2000+) CRT Projection TV'S is Sony. They have horrible picture tubes as far as construction goes. The way to know if your tube has a chance of becoming short or randomly increasing or decreasing in strenght is to open the front cover and LIGHTLY hit on the neck of the picture tube. Almost all CRT projections by Sony will have the respective color you check flash stronger when you hit it gently. I have replaced a lot of Sony tubes and never ever needed to replace hitachi, panasonic and most of other brands tubes.
The only other brands that tube can look or be defective is RCA and Samsung.
The Samsung (and some Zenith) will have the tube coolant in the Blue tube become contaminated and make the TV yellow. (bacterial or bad coolant liquid) A change of liquid and cleaning of tube solves that easily. Not a bad tube ever.
RCA RPCRT's sometimes see a tube break in half because of a surge in the high voltage due to a bad capacitor in the power supply. Again, not a bad tube but bad protection circuitry.
If the picture tubes in your Sony have ever shorted and the TV shut down, I am pretty sure it will appear in the self-test. But if, like I said, you gently tap the neck of the CRT's in the front and no flash of red, blue or green occurs, the tubes will most probably last the 15-25 years they are rated for (even at 8-10hours per day).