I'm going to posit a hard NO on this one. Time telling isn't just about getting somewhere at the same instant, it's actually synonymous with science. There are a lot of things that you can't do without accurate time telling.
Update This answer is getting a lot of flack because society did just fine without complex time-telling devices until the 1600's. Time telling is not as critical as mathematics or record keeping. From that perspective, the question should probably be "How advanced could a society get without time-keeping devices?"
To my knowledge, the critical need for time keeping was actually in tracking work hours. Realistically, you can't even get away with "you will work until I tell you you're done" with slaves. Earliest case of this was 1300BC. The Greeks and Romans both had legally defined system of seasonal hours. As such, you could say that the history of time-telling is tied to the history of labor negotiations.
You also need to ask how tightly you want to define "time keeping." Nothing tighter than hours was used before the 1500's, but there's arguments either way about how useful it would have been if they'd have had it. You also have to ask if calendaring counts as time-keeping in their view.
It's impossible to tell where you are, east-west, without knowing what time it is. Every day, the captain would spot the sun when it rose. He would compare that with the time, and that would tell him his longitude.
Baking and construction
If you can't keep track of how long your cookies are in the oven, you will burn your cookies. There's more to this, though. You need some sense of time to allow cement to properly mix without prematurely setting. The Romans couldn't have built their aqueducts without some kind of clock, or at least an hour glass. This can also be extended to brewing, soaking seeds for germination, etc., etc..
This is probably the point that your society will fall. You can't formulate the mathematics of thrown objects without tracking time, so you lose the ability to design quite a lot of simple machinery. Any civilization that tried to enforce this would be readily conquered by a civilization that didn't have such an issue with trebuchets.
Going around the issue
You can guarantee that, if you forbade them from doing so on purpose, they would do so "accidentally." This would involve building their houses in such a way that they could use the edges of it against the terrain to tell what time it was, or just parking their cart so that it would go into shadow at "closing time."
In this society, they're highly against time telling devicesWHY? $\endgroup$