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.
Be careful though: 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.
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.