Twitter wraps all links posted to Twitter with their own http://t.co shortener. This results in all clicks jumping through one additional hop (t.co) before ending up at their final destination. However, all clicks will continue to pass through any third party link shortener you are using (dlvr.it or bit.ly, for example) and you will continue to enjoy click stats from these providers.
Clicks on links posted to Twitter will follow this path: 1) first users will go through t.co, then 2) redirect to your shortened link (dlvr.it, bit.ly, etc.), then 3) pass through to your full / expanded URL.
This means your link shortening service (dlvr.it or bit.ly) will continue track and record clicks on your items posted to Twitter.
Keep in mind, when viewing your tweets at twitter.com website, your readers will continue to see links with any vanity / custom domain you are using. However, when viewing your tweets away from twitter's website -- in say their mobile reader -- readers may see a t.co link. Again, no matter what is displayed, clicks will first pass to t.co, then to your short url, then to your long / expanded URL.