When I want a multi-colored border, I stripe the dye colors I want up the sides of my disposable bag, then put in my white icing. Hope this helps.
This may not make sense or it maybe a strange method. I put a portion of the three colors individual wrap in plastic wrap. Wrap them securly an them kinda shap them like sauage. Cut the ends of the plastic wrap and put them in your decorating bag and your ready to go. Perhaps try out a small sample and see if you like the results or just simple stipe your bag with the two colors. LOL
My mom loved to make borders of two colors. She used to take a 16 or an 18 pastry bag and put two plastic pastry bags inside it. She then fed the tips of both bags through a coupler, the through the tip of the bigger bag. She then used to attach her tip and she woudl have two colors coming out!
I used to watch her do this and I NEVER understood how she managed to do this! maybe you will have better luck figuring it out! She refuses to show me again! 
They do make special bags for this.
http://www.cakecraftshoppe.com/proddetail.asp?id=1143&mcatshow=1&ncur=1
Maybe if you purchased the bags you could use them as a template for making your own.
Those specialty bags look cool! I could make some by using my Food Saver to create the seam in the middle of a disposable bag. Could even possibly make 3 compartments!
But more likely I'll use the plastic wrap trick. I love that one! I use it all the time for just single colors. Clean up is so much easierr!
I used two colors for the border of my dolphin cake. And I did it as Marissaisabel described with wrapping each color individually in saran wrap and then putting them in a decorating bag with the openings going into the coupler. Squeeze out some thru your decorating tip till all 3 colors are coming out. Works great.
MillyCakes
Your moms method works well, this is similar to what I do with the plastic wrap. I just tried that. It is better. Easier!! FASTER!! My method sometimes the icing would explode from the top and the colors rarely blended togther. GOOD TIP!!! LOVED IT. I just had to give it a try. 
Well thanks! That is my mom - can't cook toast but boy she can bake and decorate 6 cakes, bake 10 loaves of bread, and whip up some sweet rolls all while we waiting for the delivery man with dinner!! Gotta love her! 
For any of you interested in getting the bags with two chambers - they're really good to use when you don't want the colors to blend, but they are a pain in the butt to clean!!
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