The Great Firewall of China is not that much of an issue for most Chinese internet users because (a) they're not, mostly, looking for sensitive political material online; (b) most people don't speak English, so overseas sites are automatically less attractive, and (c) there are native Chinese equivalents -- okay, clones -- of blocked foreign sites. Facebook is blocked, but there's still Renren and Xiaonei. Twitter is blocked, but there's Sina Weibo - which is in many respects a better product. Youtube is blocked, but there's Youku. Google is around, but Baidu has better results for Chinese bulletin boards. And so on.
People are aware of the censorship, but they tend to identify it with site administrators (who are ultimately the ones responsible for deciding what does or doesn't get posted in discussion forums), hence a habit of sneering at "guanliyuan" ("mods"), but generally not at the government. It's not that people are stupid or unsubtle; it's that there's not much point in getting angry at the government.

I'm not so sure about the reports of people's phones cutting out. There's definitely been a radical increase in filtering and censorship here over the past month, but I'm pretty sure I've said "protest" multiple times in both English and Chinese on my (Beijing Mobile) phone without having anything happen. Speech recognition just isn't that good, unless the technology has gotten a lot better in secret -- particularly for dealing with a language like Mandarin, which is much richer in homophones than English is, and also has plenty of regional accents that would be even harder for computers to deal with.

That's not to say it's impossible -- I have no reason to believe the NYT is lying, though their China journalism is not always good -- but if it's happening, my guess is that it's limited to a small number of people whose phones are being monitored by human beings.