Bicycles may seem simple but they actually require some very sophisticated techniques to build and perhaps most particularly to use effectively. These seem to be the main reason why they were invented so late.

The histories of bicycle generally begin from Baron Karl von Drais's "draisine" (1817), possibly as a response to death of horses by crop failures caused by volcanic eruption of Tambora in 1815; however, there are some claims that push the invention of the first "running machine" back to 1792 or even 1534. These were foot-pushed frames with two wheels; cranks and pedals had to wait until 1863. Beyond some fashionable dandies, these "hobby horses" never saw widespread use, partly because they were often banned in cities and there weren't that many other opportunities where to "bike." In other words, the invention was not very useful.

Now, ancient cartwrights would've been able to produce machines broadly similar to these early running machines. It is of course even possible that some did, but we have no surviving evidence of that. It seems that outside very few cities and places with good roads, there simply weren't many places where such contraptions would be anything but interesting toys. Now, this leaves open the question, how about Rome? The most likely answer is probably again that there was no demand, as the poor couldn't afford such a machine with very limited usability and the rich had no need for one, having horses and sedan chairs.

Now, let's get back to "modern" bicycle. Since the 1820s, there were numerous attempts to build a human-powered machine - usually three- or four-wheeler for balance - using all sorts of mechanisms from hand cranks to treadles (i.e. kind of like transmission in a steam locomotive) and whatnot. I would argue that these mechanisms were possible, somewhat affordable and not exceedingly heavy largely because of continuously improving metal-working capabilities of the period. Many of the parts used in these early bicycles might have been possible for ancient blacksmiths to build... but the cost would've been unreasonably high, particularly for parts that had to fit together precisely, like bearings in wheels and in pedals. Furthermore, use of cheap iron, a product of the industrial revolution, in the frames allowed for true mass production, which eventually made bikes possible to afford for those who couldn't afford a horse in the first place.

If we now take a careful look at the bicycle as it was in around 1900 (when bicycles began to be affordable to "normal" people), we can see all sorts of exceedingly sophisticated technical details. Even if we leave aside the wonders of chemistry that permitted durable pneumatic tires, almost every part in a bicycle from 1900 is a product of refined metalworking technology. My favorite is probably the chain drive: before the industrial revolution, manufacturing the humble bicycle chain to tolerances required for smooth operation would've required probably weeks if not months of dedicated effort from extremely talented, practically artistic blacksmith - and don't get me started on ball bearings.

I'd hazard a guess that building a somewhat modern bicycle in, say, 1500s, might have been technically possible - but it would've cost at least as much as a good suit of armour, which we know tended to be expensive! And since the roads at the time were poor or nonexistent, where would the biker use the bike, even if he got one?

Key source here is Bijker's Of bicycles, bakelites and bulbs (1995).