There are only two cases where you should expect rated bandwidth:
1. 10x probe reading a low impedance circuit; short lead lengths (normal size ground clip; or even, no clip and probe tip at all, using a socketed connection instead),
2. 50 ohm matched generator and terminated scope (50 ohm internal termination enabled, or using a tee and terminator at the scope input).
Anything else, do your due dilligence.
As noted, 1x probes are quite slow, and add a lot of capacitance to the circuit. A 10x probe has less capacitance, but it will easily disturb precision, high impedance, or tuned RF circuits. A 500 ohm 'no probe' (450 ohms + 50 ohm terminated cable, reads 10x) can have pretty good response, but loads the circuit much more at lower frequencies.
The most important takeaway is perhaps: reciprocity. There is no ideal probe. We can approximate certain characteristics (like high DC resistance, or wide bandwidth), but there are fundamental limits on what equivalent capacitance, bandwidth, voltage range, noise floor, etc. can be had. There is always necessarily a loading effect on the circuit-under-test, which is to say, a probe necessarily exhibits an RLC equivalent circuit -- it's never an ideal open circuit (infinite impedance). Keep this in mind when performing measurements: rather than trying to avoid those properties (e.g., using a low capacitance FET probe to measure logic signals), try to work with them (e.g., using a probe of known capacitance to measure logic signals -- a capacitance which is representative of a real circuit using those signals!).
And reciprocity takes you quite some distance in general, because all passive circuits interact in both directions. You can't build an RLC filter without it! Amplifiers (and a few other peculiar components) are actually the odd ones out, because they can disobey reciprocity: having gain for one thing, but also having unequal forward and backward gains.
Tim
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