A meter would tell you if it's a matter of too much signal to the output tubes.
How would you hook it up and test?
The first fact to grasp: If the grid is driven momentarily more positive than 0v, the grid draws current and can cause distortion of both the output tube, and the phase inverter. The phase inverter gets distorted because the grid no longer looks like a ~infinite impedance, and becomes a heavy load. That drags down the gain of the PI, while at the same time causing it to distort.
Additionally, the grid current can cause blocking distortion by placing a charge on the coupling cap between the PI and output tube, which then causes the grid to latch at a certain voltage despite a changing input signal. Cause all kinds of weirdness.
So we might play most guitar amps in a manner that drives the output tubes to distortion, but we simplify design by assuming the grid will never be driven positive. "Driven positive" means a positive incoming signal is more than the value of the bias (ex: -34v bias, 38v peak input signal).
So measure the bias voltage. Then, measure the signal at the output tube grid while injecting a test signal which approximates a guitar signal's strength. The test signal will usually be a sine wave, and a guitar doesn't produce sine waves. So, you would likely want to take a guitar and measure it's output with you banging on it, while the meter is set for a "Max Hold" function. You will want to know if your meter performs this reading based on maximum RMS signal or maximum peak voltage.
Depending on which it is, you apply a test signal with a signal generator that matches this peak
at the input jack. You may have to convert a measured RMS voltage to equivalent peak voltage or vice versa. Checking the peak guitar signal using a scope may be a good idea if your meter's peak hold function measures RMS or average voltage (because the guitar's non-sine nature will make the meter lie to you).
Once you've established an appropriate test signal based on the guitar/pickup typically used, you can measure the value of the voltage at the output tube grid (converting RMS or average to peak, if necessary) to see if your typical settings of volume, tone, etc. drive the output tube excessively. Obviously, you can fine tube with a single-coil Strat and have an amp that doesn't sound right with a humbucker guitar, depending on how far on the edge you get (or how unwilling the player is to turn down until the amp sounds right).
Ken Fischer suggested that many great amps (likely assuming you don't need a master-volume gain machine) are designed such that the output tubes start distorting first, the phase inverter shortly after, then the other preamp tubes. It's part of what makes an amp touch-sensitive, where soft playing is largely cleanish, and a heavy attack is distorted.