Danielle's right - what one person thinks is too much noise another may find perfectly acceptable. It's really up to you to determine just how offended you are by noise in a shot - some folks don't mind a little graininess if the details are all there and they get the shot...others want 100% clean and free of any noise whatsoever.
Then there are other variables - such as, reducing noise via noise reduction software...which can make photos less noisy, or even noise-free (if willing to sacrifice some detail). How much noise reduction to use if any is all dependent on your own taste and what viewing size you'll be using for the photo or print.
And then there's the exposure. Noise is worse when the exposure is too dark - underexpose a shot, and you'll get worse noise. What ISO you can use on any given camera is partially dependent on how well you nail the exposure - a nicely exposed ISO3200 shot could be full of lovely detail and minimal noise, while another ISO3200 shot with the very same camera that was underexposed can be an awful mess of noise and smeared details.
See these examples...both taken with the same camera and both unprocessed. First, an underexposed shot at ISO1600:
ISO3200 should be noisier than ISO1600, right? Well, here's an ISo3200 shot that was properly exposed:
Note the ISO3200 shot actually has less noise and better details than the ISO1600 shot. That's all because the exposure was better in the second shot. Sometimes, raising the ISO one stop to get a better exposure can result in a better, less noisy photo than using the lower ISO and underexposing the shot. Cameras are different, and some might not show as much variance...and some cameras can go to much higher ISOs and still be mostly noise-free...but how much noise you consider too much is really a personal thing!