My levels check out accurate.
I think the point you are missing, and that RKC and others are alluding to, is that while your carpenter's levels may be "accurate," they may still be not precise enough for leveling a pool table.
Speaking as an engineer (not a table mechanic), there's an expression that goes "precision is repeatability; accuracy is truth."
One might also use the term "sensitivity" to refer to precision in measurement.
Your carpenter's level may be such that you can't visually detect a displacement of the bubble between the center index marks over a range that is easily detected on a precision level (referred to variously as machinist's level, millwright's level, precision level). And, that amount may indeed make a difference on a pool table.
You might think that you can measure with the carpenter's level more precisely by being very careful in observing the bubble position, and by reversing it to take out any error in the level's "zero." But you can't. The primary reason for this is that the vial that contains the oil and bubble is manufactured with an arc to allow the bubble to move when the level is tilted. This arc, for carpenter's levels, is on the order of 8" radius or so. A really, really good carpenter's level might have a radius of 18" This is still at least ten times less than the vial curvature of a machinist's level. The machinist's level will indicate a deviation from level that you will never see with even a very good carpenter's level.
So, while your level may indeed be accurate within the precision of the instrument, it's not precise enough for this kind of work.
An example that might help is that of a Voltmeter. Say you have a meter with three digits, and on the hundred Volt DC scale it reads from 00.0 to 99.0 V. Let's say it's been recently calibrated, too. You probe a node and it reads 45.0 V. We then get a more precise, recently calibrated meter and the same node reads 45.049V. Note that both are accurate. The latter is more precise. And, the latter is necessary if you need precision down to the millivolt (thousandth of a Volt).