How precise are altimeters?
While Accuracy and Precision are closely related, they're not the same thing.
In the context of your question both are important:

Accuracy

In the US altimeters are broadly required to comply with FAR 43 Appendix E limits for accuracy (technically this is regulatory for IFR flight, but TSO'd "sensitive altimeters" will generally meet these requirements - at least when they left the factory - and that's what you're going to find in most aircraft).

The accuracy requirements provided in FAR 43 are based on mechanical altimeters, and are subject to a number of requirements (friction error limits, hysterisis limits, etc.), but the test tables there are a good guide to the accuracy of "typical" altimeters.

Also note that the accuracy of sensitive altimeters depends on the pressure setting the pilot enters (an incorrect pressure setting could mean substantial error). In "the flight levels" everyone uses the same standard altimeter setting (29.92 inches of mercury / 1013 millibars) to avoid this issue (similarly the ATC transponder transmits an altitude assuming a pressure of 29.92 inches of mercury - ATC's computers correct that value based on local altimeter settings as required).


  Depending on the design digital altimeters may also be immune to some of the accuracy-affecting issues in an analog/mechanical altimeter - for example a solid-state digital altimeter should not suffer from "friction error" as there's no rotating/moving shaft involved.




Precision

Analog altimeters are "reasonably precise" -- take this example:

The tick marks are every 20 feet. One could easily eyeball 20 foot increments, and even 10 foot increments (with the pointer directly between two ticks). Similarly the pressure settings are marked every 0.02 inches of mercury (or 1mb), allowing for a fairly precise setting of the value.  

Digital altimeters are extremely precise (often offering one-foot precision in their display, and 0.01 inches of mercury for the pressure setting).


  I believe the precision requirements for altimeter markings are specified in the relevant TSOs, but I don't have those documents handy.




In both cases the precision of the markings can exceed the accuracy of the device: At 10,000 feet the FARs allow an 80 foot tolerance, and whether your measurement is presented as a needle position (roughly to the nearest 10 feet) or a number (to the nearest foot) you could still be 79 feet higher or lower than you "think" you are and be "within tolerance".

So, combining all of this together - If we have two aircraft flying along indicating the same altitude (say 10,000 feet) and both altimeters are perfect there will be zero feet of vertical separation between them (the static ports of each aircraft will be at the exact same altitude).
On the other hand if both aircraft are at the outer edge of tolerance (one +80ft, one -80ft) there will be 160 feet of vertical separation between them (discounting visual "slop" in the analog altimeter and assuming the pointer is perfectly on the line and the pressure setting is perfectly dialed in).

(In either case the answer would be "WAY TOO CLOSE!")