Bear in mind that what you're seeing is the behaviour of an oscillator when the extra capacitance of a scope probe is added. I tend to avoid designing with simple crystals wherever possible, because they can be rather sensitive and unpredictable; an integrated crystal oscillator module is more expensive but virtually guaranteed to be reliable - and because the output is buffered, you can probe it with a scope without affecting its frequency.
Borderline cases like these where some crystals don't oscillate at all, or at the correct frequency, probably indicate that the crystal, the capacitors and the drive level from the logic inverter aren't properly matched. It's a design issue that should be checked by the engineer who specified the particular components, and in particular, someone needs to check that the specific parts called up by the engineer haven't been substituted for apparently equivalent ones by someone unqualified to make that decision.
Sometimes it can help to probe other signals that should be derived from the master clock, rather than trying to probe the crystal itself. If, for example, a microprocessor has a PLL in it which uses the crystal as its reference, the outputs of the PLL will give a strong clue as to whether or not the master clock is OK. If, say, there's a 33MHz input from a crystal and a 100 MHz output from the CPU going to the DRAM, then if that 100 MHz clock is OK then the reference crystal probably is too.