multimeter
I found the following note with a set of test leads we use for our Agilent 34461A[1][2]:
This Test Lead Set is designed for use with the Agilent 34xxxA Multimeters, and also the U274A, U3401A, and U3402A Multimeters. Do not use with the Agilent 3458A.
I suppose they're simply not calibrated for use with other meters and 'too inferior' for use with the 3458A[3]. But afar from precision, is there any reason not to use them on a different multimeter, provided they fit?
In short, yes.
The way a digital multimeter measures current (not a loop current meter) is that it measures the voltage drop across a small precision resistor. For reliability, simplicity, and repeatability, no switches or contacts are placed in series with this shunt resistor. Some meters will have a fast-acting fuse to protect the shunt resistor. Basically, the switches will control the electronics hooked up to the shunt resistor, but they won't disconnect it.
You can perform an experiment with your meter as some degree of proof that this is true. If you configure your meter to read resistance, put the positive probe into the positive current probe socket. You should measure a very small resistance. This shows that the meter doesn't disconnect the shunt resistor while in a separate mode. It would be a small leap of logic to say that if it didn't switch off the shunt resistor then, it probably couldn't when the meter is off.
Alternatively, you could measure the current using a second meter.
The display is not the limiting factor (as you point out). Fundamentally, there is a cut-off. If the display was the most expensive part in the meter, then most meters would probably read up to 9999 on all digits. However, there's a couple factors that make the meter ranges what they are.
There is a limit to how high a range goes - the count is a design decision for the meter maker. A 20... meter isn't the only type available either, I personally have a 6000 count meter.
Logarithmically, you get the most bang for your buck by having 2000 counts. See the image of the number line below. By doubling the counts from 1000 to 2000, you get a bigger bang for your buck, and you also get to use one more digit (better for marketing).
There's another reason for not dropping the least significant digit. Part of the gotcha with auto-ranging meters is that you can easily forget that the meter has different measurement ranges, and that the measurement accuracy changes with each range. For example, compare the following displays:
193.00 k
.1930 M
193.0 k <----- Would you notice the missing digit as easily?
Personally, I see it as a more obvious way of conveying information about the measurement. It doesn't detract much from the readability in my opinion.
Best Answer
In addition to the excellent answer given by @JorenVaes there is also a simpler electro mechanical reason for this caution. If you take a close look at the lead connections for the 34xxxA multimeters you can see that they are made like shown here:
The above type of jacks are designed to be used with the shrouded type of banana plugs that look like these:
On the other hand the 3458A meter uses test lead connections that look like this:
As you can see the shrouded leads will not mate with the latter type of jacks. Shrouded banana plugs have come into use on many meters because of the safety they offer. The types of meters may be used in high voltage or high current applications where the user may detach the lead from the meter when it is still clipped into the test circuit.