<p>Let's say I'm modeling a simple box with a lid. Just as an example, we'll say the <strong>outer</strong> edge along the top of the box is 50 mm x 50 mm. With 3D modeling software, it's easy to build a lid for this box to surround the top with an <strong>inner</strong> edge size of also exactly 50 mm x 50 mm ...but this seems like a bad idea. Surely I'll want some kind of of gap, to ensure an easy on/off. An <em>exact</em> fit seems like it's asking for trouble.</p>
<ul>
<li>How much gap do we leave for this kind of thing?</li>
<li>Is it related to nozzle size?</li>
<li>I suppose it also matters how tightly you want to fit, though I expect in cases where a tight fit matters some kind of snap or clip would be used.</li>
<li>Are draft prints with larger layer sizes useful for figuring this, or do the rough layers make things seem tighter than they'll be in a final print?</li>
</ul>

Short version:
basically, this depends on your printer, make, model, type, state of maintenance, extruder, slicer settings, belt tension, play, friction, etc.
Long version:
Basically your printer determines how accurate it prints; you can influence the accuracy a little by calibrating and fine tuning the printer. What regularly is done is to print
calibration cubes
of fixed size. Before you do that, you should read "
How do I calibrate the extruder of my printer?
"; this explains to calibrate the extruder. With a fine tuned extruder you could print those XYZ calibration cubes, or in your case create a box of e.g. 50 x 50 x 15 mm. When you measure the length and the width with a caliper, you will know how much the tolerances are for this print size. Eventually, you could change this by re-adjusting the steps per mm in the firmware of the printer, but this is not always a recommendation (as your steps per mm should be related to the mechanical layout of the used mechanism, e.g. the belt size and pitch in combination with the pulley and the stepper resolution).
Please also look into the answer of "
How to make moving parts not stick together?
"; this answer hints to printing a tolerance calibration model that uses diabolic shapes set apart from the outer object by several values for the offset between the pieces. When you print this you can find out what sort of tolerance works for you. Please do note that the tolerances on smaller parts may be different than the tolerances on larger parts.
The answer on your question thus depends on your 3D printing machine, but usually the tolerance values range in the few tenths of a millimeter. To enable a lid on top of a box like in your example, you need to keep the tolerance in mind when designing the lid. Usually an extra few tenths of a millimeter will do the trick, but if you make some test prints first you will know exactly.
To answer the question what the influence is of layer height on tolerance, I
quote
:
Load a 25 mm cube into your slicer and set the infill to 0%,
perimeters to 1, and top solid layers to 0. You’ll also want to print
it at a fine resolution – I chose 0.15 mm and it actualy did make a
small (0.02 mm) difference in the wall thickness as opposed to 0.3 mm.
So yes the layer height has an effect, it is very little though.
An interesting read is "
A Guide to Understanding the Tolerances of Your 3D Printer
" from "
matterhackers
".
Furthermore, when you have calibrated the printer but still run into small deviations, is that most slicers will allow you to compensate for X and Y dimensions.