Yes. Get a nice following around a certain interest, and then find out how to monetize that interest.
IF you have plenty of followers...and I mean TENS OF THOUSANDS, that are all mostly in one niche, you can make good money by offering affiliate links. But do it sparingly, don't spam whatever you do! And always try to help your followers and add value.
Try going to sites like ClickChores, where people post jobs paying for Twitter followers/people who can deliver Twitter followers
There is money to make directly with Twitter really. If you mean can you utilise twitter to help you make money then the answer (simliar to most other Social Media sites) is YES. There are various ways to do this - do you sell a product online or are you an affiliate / internet marketer - then USE twitter to drive people to your blog or website, where there is valuable content that people will want to read more about HOW DO YOU DRIVE PEOPLE TO YOUR SITES? In the exact same way that I generated nearly £17K in orders for my family's offline business in 12 months I took to Social Media sites and I started communicating with people, answering questions, asking questions, tweeting about daily life, being helpful, showing my knowledge of certain areas ... And after a while I chose to tweet about what my business did, what we sold (nb - I didn't pump out tweet after tweet of offers / sales - nobody likes to be over sold do they?) It wasn't long before 1 sale became 2 sales and then I had built an online community that were fast becoming regular customers, and also recommending our services. There are a LOT of people out there who are making money by guaranteeing followers, and paying you to like them, but really, ask yourself this. Are you serious about using Social Media to build a sound reputable business? If you are then there really is a right way to do it, and for it to be a lasting business the foundations need to be right. The trick to generating sales and customers is to be a person, with a personality that cares about those they choose to follow.
Many people make money by using twitter.... Just offer a product for sale online and provide a link in your tweets for followers to click on.
SponsoredTweets has been by far the best for me. You nee a lot of followers though. I have about 10,000 niche followers and I made $50 last summer. MyLikes is also decent, but it's dependent on how many clicks you get, so it only works when an ad matches perfectly with your niche. I just signed up for Staree.com and it has been the worst in terms of revenue. However, it's a great platform for posting pictures (sort of like TwitPic), so I view the money as more an added benefit.