In my experience, many NATs just drop UDP packets altogether, but still allow TCP through.
Similarly, I find the Internet at my local Starbucks to be some of the fastest, most-reliable networks around; even faster than some local ISPs hooking up to my home.
Note that any NAT that drops UDP packets altogether will basically disable DNS and VOIP type applications or at least degrade the experience in serious ways. I haven't come across many such NATs recently.
Me neither. I have come across networks that filter everything but a few TCP ports (like 80 and 443), but that's a matter of draconian firewall policy rather than a NAT limitation.