Maybe this is not directly related to W3 Total Cache but I'm reporting here as well. It might be useful for someone else.
All my websites are hosted by DreamHost. Almost all are WordPress, and most of those use W3 Total Cache. When I was moved from a failing server to a 'new' one, for some reason, gzip compression was not working properly: depending on the browser/operating system, users would get a compressed file to download (usually called download.gz or domain-name.gz). This usually means that the wrong headers are being sent. When logging in as a user — which will bypass W3TC — the website was evidently working well. Cleaning cache helped for a while... but the problem happened over and over again.
I first thought that only the websites with most traffic had a problem, because I have CloudFlare on top of them, to improve performance (CloudFlare is a free CDN-like, cloud-based service, which will take all your mostly static content and spread it around 14 datacentres in the world; it will also screen out web attacks from hackers, add IPv6 if your hosting provider doesn't support it, and even do SSL for non-SSL websites). CloudFlare also does gzip compression, so I guess that something had changed and now Apache (?), W3TC, and CloudFlare were all gzip-compressing the very same pages. Beyond the waste of CPU, it meant that at some point something went wrong.
Well, I turned gzip compression off on W3TC. Then I subsequently noticed that even those websites with W3TC but not CloudFlare had the same problem as well. So I turned off gzip compression on all websites using W3TC!
This seems to worked out fine! No more problems and the websites are actually faster than before. If my assumptions are correct, than it's not surprising: each successive gzip compression requires CPU.
Now in my case this means that only the websites going through CloudFlare will be properly compressed, and all others using W3TC will have no compression. I really don't know if there is a bug on W3TC or if it's some kind of slight incompatibility between W3TC and my hosting provider...