My opinion is that no fret buzz is acceptable with ordinary picking or strumming. I won't rehash what others have said here, especially since MGM covered the technical aspects far better than I can.
The catch, of course, is that "ordinary" picking or strumming varies from person to person. So, sometimes, you have to look at playing style as a cause of buzzing. Put quite simply, some people can successfully play with very low actions while other people playing that same uke will buzz like crazy. I learned this the hard way when I had a custom SG guitar built for me by a local builder who has since become a bit famous. Anyway, the afternoon I picked up the guitar from his shop, he was playing ZZ top covers on it, etc., without a hint of buzz even though the action was crazy low. The moment I touched it it became a buzz-o-matic. We ended up raising the action at the 12th fret by a good 2mm so I could play it without buzzing. However, I learned my lesson from that and began working on technique, and over a period of a few weeks was able to lower the action back down to where he'd had it.
If you push the strings down or pull them up as you play they are going to vibrate in a different plane that is more perpendicular to the fretboard and thus brings the strings closer to the frets at one end of their cycle. (Actually, they don't really vibrate in a plane, more like an ellipse, but you get the idea.) All other things being equal, if you brush the strings more parallel with the fretboard you will be able to play lower actions without buzzing.
The other technique-related thing is how hard you strum or pick. Good players strive to achieve a variety of "dynamics" throughout the song, emphasizing certain phrases by increasing volume, etc. If you start out with a hard strum as a baseline then you are going to have to hammer the instrument really hard to get a louder dynamic, and that's going to put you into buzzing territory. It's best to learn to start with a softer "soft" dynamic - if you need more volume to be heard in a band mix then get a better uke or use amplification.
I'm still a fairly "enthusiastic" player yet I've found that I can play ukes with ridiculously low action if they've been set up right with nice level frets. I got my Kiwaya longneck soprano from MGM and the action at the 12th fret is almost ridiculously low, yet I have to hammer it pretty hard to get a buzz. I compensated the saddle that is on my KoAloha longneck and that brought the action down to the same territory. I don't recall what they measure out at, but it's lower than I've seen on any body's uke and they don't buzz. Now, if I was to hammer them, or hit those strings vertically, yah, it would probably buzz.
Bottom line, the action (especially at the nut) needs to be low enough for good intonation - beyond that you should personalize the action to your style. If you've got a smooth delicate touch - or wish to develop one, then gradually bring the action down at the saddle. If you're "enthusiastic," and happy being so, then leave the action at the bridge higher.
John