Molly:
Many printing devices ignore the overprinting attribute, probably so the manufacturer's tech support won't be flooded with calls from angry customers about white text they see on their monitor not showing up on their prints. Fortunately, Adobe provides a workaround for this that you may be able to use to solve your problem. When you print (it has to be a composite print), choose overprints:simulate in the advanced category. When you use this option, overprinting is not used, but the objects are changed to achieve the same result. E.g., a 0/0/0/100 object overprinting a 100/0/0/0 object is changed to 100/0/0/100. If you have spot inks, there will still be overprinting in the postscript output because there is no way to simulate spot ink overprints. Depending upon the inks involved, you might be able to convert any spot inks to a process primary to get around this. If you must print separations, you could print composite to "Adobe PDF..." or postscript and then print separations from the resulting PDF out of Acrobat. You could also probably reopen the PDF with Illustrator, but check it carefully for any goofiness, and don't tell Dov or Leonard.
Just in case, make sure the document color mode is CMYK.
An overprinting object always knocks out inks that it uses, no matter the percentage, so sometimes setting the blending mode to multiply yields a better result. For example, if you have an object that is 50% cyan and 100% magenta on top of an object that is 100% cyan and 50% magenta, the trap color should be 100% cyan and 100% magenta. If you add an overprinting stroke to the top object with the same color as the top object, the overprinting attribute will make no difference, and your trap color will be 50% cyan and 100% magenta, the same as the top object, so your stroke would effectively just make the top object a little larger. The cyan wouldn't be 100% because the stroke contains cyan, so it knocks out the cyan underneath and you get the cyan that the stroke contains - 50%. In this case, a multiply blending mode would yield the correct result. This probably isn't your problem because you mentioned it looks right with overprint preview turned on, so I'm guessing all of the trapping strokes you've made are on clean breaks from one ink to another.
Even if there are no shared inks between the adjacent objects you're trapping, there is still a way you could be suffering from this "shared ink" problem. If the value of an ink is greater than zero but less than the threshold that normally gets rounded to zero (about 0.196 percent), it could be interpreted one way by Illustrator and another way by your RIP or another device. Color values are often quantized to one of 256 values. Illustrator appears to perform this quantization when previewing overprints. If you create a box that is 100% cyan, then a box that is 100% magenta set to overprint and placed on top of the first box, the solid cyan should show through. You can add .19% cyan to the top box and it still shows the solid cyan underneath. If you increment it to .2% cyan, then it should knock out.
If you happen to have a shared ink in the overprinting strokes that is quantized to zero by Illustrator but not by your RIP, you will get different results from each. I have seen objects in artwork created by Illustrator that were 0.1% black in a PDF or EPS file, but read exactly 0% in Illustrator when it opened the file, even though the color palette can show hundredths of a percent. To test for this bug, you can use a tool like Pitstop or save an EPS file containing just one offending object and examine it with a text editor to see what the color is recorded as.
You might try placing the trapping strokes on their own layer(s), then printing separations individually. You would want to turn off layers containing trapping strokes that are not in a particular ink color when printing the separation for that ink.
As a last resort, you could rasterize the artwork with Photoshop at 600 ppi without anti-aliasing, and use Photoshop's trapping function. If you have spot colors, you would have to output separations, rasterize them separately, then combine them (Don't cheat and use process colors as proxies unless the density differences are similar).
You can often apply trapping automatically using Effect > Pathfinder > Trap.... You need to group everything and then apply the effect to the group. Then choose Object > Expand Appearance if you want to release the effect and get separate editable objects for the trapped areas. This can get too complex very quickly and you may need to subdivide the artwork into separate groups for it to work. If you have spot inks, overprinting will still be used, but I don't think the effect uses overprinting for process colors.
