Tomatoes with wilted stems

Are the tops - above the wilted stems - still green? How much of the stem is wilted?

It may be mechanical damage. You must be careful as you guide the stems around the string, that you do not twist the stems.

If the plant above the wilted stem is brown, you might have a salt problem - try watering very heavily two or three times without any fertilizer being applied.

Is there any evidence of cutting, chewing, or boring on the wilted stems? There is also the possibility of insect damage. The tomato horn-worm usually attacks the leaves, rather than the stem, so that doesn't sound like it.