Does tube testers tell how much life is left for tubes?

In the end, we usually are alerted to the possibility that a tube or tubes are failing by the fact that the our system sounds "off", not as good as usual.  Only then are we motivated to test tubes.  At that point, if any tubes measure low in transconductance, I ditch them in favor of new.  That nearly always results in restoration of SQ.  I own an ancient Hickok 533A, which I can use to test small signal dual triodes only, to include octal based tubes like the 6SN7 and 6SL7, and it is not much good for testing modern very high transconductance triodes like the 5687 group or the 6DJ8 group, because of its current limitations.  I just shoot for an approximation of "good" with those.  The 533A is no good at all for testing the monster 7241 triodes in my power amplifiers, and for that I just go by listening and then measuring the tubes in situ.  The 7241 is a US milspec power triode developed for use in radar installations.  It can easily run on up to 1A plate current, although I do not stress them that much.

The best measure of the condition of the tube is to measure transconductance.  Transconductance is a measure of the change in current flow across the tube ("plate current") for a given change in grid bias.  (I didn't look that up, so if I got it wrong, mea culpa.) Since it is a current divided by a voltage, its units are inverse to ohms, called "micromhos".  So far as I know, the only vintage tube testers that measure transconductance are Hickok brand and a few other brands that were actually made by Hickok for that brand.  So, you start with a table of transconductance readings for the particular tube under test, and you use the tester to find out whether that tube sample meets spec.  Typical Hickok testers have a big analog meter on the front panel and will indicate to the user whether the tube needs replacement if it is too far under the standard transconductance for that tube type.  Many times a perfectly good usable tube will measure a little below max transconductance.  When the tube is really at death's door, it will look to be very low in transconductance.  Another way to test a tube is simply to measure current and voltage while it is in the circuit where you are using it.  That's actually better than using some ancient uncalibrated tester.  The old testers almost never can test a tube under the actual conditions of its use in circuit, because they usually cannot develop adequate plate voltage or current for high current tubes to work best.