Is it possible to sabotage a horse-drawn vehicle in such a way that it fails a bit after leaving town? If so, how precise can this delay be and how would it be done?
Compromise the linchpin.
Carriages often used linchpins to hold the wheel to the axle. A failure or loss of that pin would lead to the wheel falling off the carriage, with consequences that would depend a lot on the specifics of the carriage and the wheel—a relatively stable vehicle could simply end up effectively immobilized for a time, while a less stable vehicle could fall on its side at speed and be catastrophically damaged.
How predictable this is depends on a lot of factors; however, one option would be to replace it with something very weak (a dark wax would probably not be noticed on a cursory inspection) on a wheel that you know will be on the inside on a sharp turn heading out of town. Most likely, the wheel would stay on the axle until that turn, when the lateral forces would cause the wax to give way and the wheel to fall off. For a yet-faster failure, remove the linchpin entirely; for a longer delay, use something too weak to do the job but strong enough to last for a little while (eg, replace an iron linchpin with dark-painted pine). The longer the delay, though, the less predictable the failure.
One thing about this method is that, depending on the results of the accident, it might never be discovered as sabotage. If the carriage were basically destroyed, it's plausible that anybody investigating would simply say, "oh, the linchpin broke, what a tragic accident," and not go digging into the axle enough to realize that the "pin" was a fake. If the carriage were intact, though, the ruse would almost certainly be discovered when somebody went to mount a new wheel.
(To give credit where it's due, the idea of a wax linchpin comes to me from the book "The King Must Die", in which the linchpin of Theseus's chariot is replaced by wax, so that the wheel falls off as he rides into battle.)
The wooden axle can get a V cut of suitable depth and over time it will break, due to the forces given by the road bumps and the carriage load. I don't think this method can be very precise, as it depends on the conditions under which the carriage is operated.
Alternatively, you can sabotage the brake: carriage had a lever operated brake to assist going down steep roads, usually consisting or in a wooden pad directly pushing on the wheel surface or a tensionable leather belt wrapped around the axle.
With the sabotaged brake you are sure the failure will happen at the first steep descent.
Last but not least, you can poison the horses' food, and that will take effect after a certain time from the last time they ate.
How about something reliable, reasonably timeable, not at all obvious, and using materials available in the timeperiod.
Just rub some Oxalic acid crystals into the grease on the coach's axles.
The stuff is quite inert, looks like just some yellow sand/dust, and will sit there completely inert until the coach starts moving at speed. You can apply the acid overnight, it remains to be just a dust covering on the axle until worked in to the grease heated by motion and heated by normal operating friction.
Once the stagecoach is moving, the acid crystals get worked into the grease, warm up, and promptly start eating away at the smooth surfaces of the loadbearing axle metal. Within an hour of staring to move, but no less than about 20 minutes, the coach has rough axles with zero functional grease on them. The coach either brakes to a halt with wheels that refuse to turn, or if the horses are strong the axle overheats, further activating the acid, and completely snaps off.
You have an immobile coach with a simple, obvious case of axle failure. And no obviously visible signs of tampering.
Maybe far-fetched, requiring some installation but the technology would have existed.
install a gear driven device that advances a piece containing a saw or blade towards the axle that is driven as the axle turns. As the carriage travels, the axle turns, which in turns spins a gear which then turns a worm gear that moves the blade forward.
You can have a ratio of 100 turns of the axle moves the blade 1cm. After 1000 turns the blade could travel 10cm. For a 1m diameter wheel, that means the carriage would have traveled about 3km. If this device is geared and sized right and installed correctly, you could theoretically make this happen.
The math to figure this out, the technology was there and material components have existed for centuries. It just would be a very highly precise piece of machinery, for the time, that would be expensive and not really available to the public.
Medieval car bomb
Plant a small keg of gunpowder somewhere on the cart where it won't be noticed, with a slow-burning fuse. These can be calibrated to burn down in a predictable length of time, up to a certain degree of accuracy.
Alternatively, for a more low-tech no-preparation-required alternative, if the cart has a load of goods packed in straw, simply toss a hot coal from a fireplace or blacksmith's forge into the straw. It can smolder for a while before the straw catches fire, and when it does, it may be too late to stop it, especially if there's no water nearby.
If you want to control the exact distance traveled before the breakdown, you might partially cut the axle and mount a coil of rope under the carriage with one end attached to the carriage body and the other to the axle.
When the wheel turns, it coils the rope around the axle and when the rope runs out, it pulls on the compromised axle to break it. Another way is to attach the wagon body side of the rope to a sabotaged fastener under the wagon so the rope pulls the fastener out, causing the sabotaged part to fall off.
You might need to use some thread for the first part of the rope, since you would need about one tenth of the distance-to-failure in thread or rope. So, a few dozen feet of rope attached to the body, with miles of thread attached to the axis.
I'm surprised that I haven't seen a this as an answer yet; compromise (using a cut partway through) the pole (the bar that keeps the horses equidistant), or the traces (the arms attached to the horse's breeching).
I'd imagine the failure would be certain, though the timing would not be precise.
A rasp on the axis, with a weighted block below
Carriage axes were made of wood.
Use a wooden block with a hole for the axis. Actually not a hole, it should be open at the top - a short piece of rasp blade is used to close it.
Make the block just heavy enough that the rasp will eat enough through the axis to break it. "Soon after out of town" sounds reasonable if the rasp is of good quality.
Downside: This will be noisy. A seasoned, attentive coachman may notice because he tends to know how it sounds normally. It might be advisable to make him unavailable, or drowsy, or bored, or have an utterly disinterested one, to make the plan work.
Whatever the saboteurs do, they need to test how fast their mechanism will work, since there's no way to predict that, given that carriages were crafted, not mass-produced, so the axis and other elements would vary slightly and also have pretty different stability depending on how well-maintained they were.