Just because a calibration meter makes a reading at a very low light level, even a consistent reading, that doesn't mean the reading is accurate. Often specs are stated in such a way that low light readings can be a factor of 10 "off" from accurate. These are colorimeters, not light meters. They are measuring red, green, and blue light. When you measure .001 fL for a black level, you are measuring blue like at a level of .00007 fL which is HIGHLY unlikely to be possible. Red light will be measured at a level of .00021 fL and green will be at .00072 (approximate values, and assumes black is "neutral" and does not have an excess of either of the 3 primaries. Photometers can measure light at much lower levels.
Any by the way, when you are measuring light levels as low as the ones quoted in this thread, if you are measuring the black level in a basement at midnight with no moon, blocking the window with a light-proof covering should change the reading. A single LED illuminated in the room will change the reading you are getting even if it is not in the meter's field of view (assuming the meter is at least a few feet from the measurement surface).
Unless the meter has been verified with a highly accurate light meter (that likely will not read in the same units as the colorimeter, so the measurements will have to be converted from the photometer's units to fL to make a valid comparison), I wouldn't trust any readings lower than 10x the meter's lowest specified measurement level. So if the meter specs say .001, it is probably trustworthy to .01 unless it has been verified at lower light levels with an accurate meter capable of measuring very low light levels.