Yes, you can use both a browser with a direct connection to the internet and a browser that uses the Tor proxy to access the internet (such as the Tor Browser Bundle), and still have the anonymity benefits of Tor when using the right browser.
There are two risks I am aware of. First and most significantly: modification of pages you are viewing to expose your identity if your sensitive browser holds identifying cookies. Secondly you need to be careful to avoid helping passive attackers carry out timing attacks.
Schneier believes the NSA's QUANTUMCOOKIE program may modify a (sensitive) page you are viewing over Tor, to inject a part of another website that will trigger your browser to send identifying cookies:
My guess is that the NSA uses frame injection to surreptitiously
force anonymous users to visit common sites like Google and Facebook
and reveal their identifying cookies.
They could do this through compromising / owning the exit node, or one of their other internet-scale programs to modify traffic between the exit node and the intended server.
Of course, this highlights a generic attack that isn't limited to the NSA.
It's not completely clear from the question whether you would be exposed to this, but the best mitigation would be to never use your sensitive browser for purposes that could identify you, and ensure it doesn't store cookies over sessions. To be specific: use a unique browser profile (or TBB install) for each identity you wish to have, and don't mix them. In the simple case of sensitive vs unsensitive, use a single browser for sensitive use only, and another for unsensitive, potentially identifying use. Both browsers can be used at the same time, provided you don't mix what you use them for.
If your browsing on the directly connected browser is related to your browsing over Tor, that could assist an adversary in carrying out timing-based confirmation attacks.
For example, you're at an internet cafe with your Tails distro, and busy chatting anonymously over Tor to a reporter at the Guardian, whilst at the same time browsing outside of Tor the Guardian's coverage of your previous story and researching asylum in Ecuador... the government agency notices your interesting non-private browsing and the fact that you are using Tor, and can make some intelligent deductions about what end points you might be communicating with.
Having dramatically narrowed down the possible end points you might be communicating with from "the internet" to "places of interest to people with a reason to flee to Ecuador and interest in The Guardian", their confirmation job becomes quite a lot simpler.
Of course, if you're looking at something with wider appeal over a direct connection whilst doing your sensitive browsing over Tor, then whilst you arouse suspicion for using Tor, your well-resourced adversary doesn't have a lot more to go on, so carrying out confirmation attacks will be a lot more expensive for them.
Have a unique browser per identity you want to be kept separate, keep anything remotely related to an identity within the right browser. Oh, and don't accidentally copy and paste a URL / search term / email into the wrong browser.
Lucas' answer rightly points out that if the NSA or equivalently well funded adversary is specifically trying to track your activities as a high priority target, and not just blanket monitoring all Tor users, for example, then this question is fairly moot.
Update: added cookie extraction via frame injection