You probably have variations or different shades of the same color in the design. Make sure you select all objects and create a selection pallet so you can look at all the colors associated with the design. If you have like colors try grouping the objects and converting them to the same color. Also make sure you convert all of the colors to Pantone Colors using the built-in pantone solid coated book in illustrator. That will tell the screen printer exactly what color ink to print.
Also, you may want to look into process printing or overprinting. In my experience, CorelDRAW does a much better job in preparing designs for screen printing than illustrator does. I use a combination of CorelDRAW X8, Advanced t-shirts SimpleSeps4, and Accurip to prepare my films for screen. I can live preview my separations before sending them out to the rip. It gives me full control over the halftones, LPI, choke, etc. And I can run the seps 100 times until I get it right on the computer without wasting a drop of ink or single a sheet of transparency film.
Advanced t-shirts also has a neat program called SimpleSeps Overprint which can create a 3 screen design and still lay down an entire gamut of color. With overprinting you use halftones and color blending to simulate a wide range of colors, a very useful tool in the printing industry. Check out these videos for more info: