Why aren't bipolar patients supposed to take antidepressants?

Complications. Antidepressants may provoke a switch into mania in bipolar patients. That risk may be reduced with a mood stabilizer. For many bipolar depressed patients, treatment with a mood stabilizer alone is more helpful than combination therapy which includes an antidepressant.
They can. They can with caution & a mood stabilizer and/ or an antipsychotic on board--otherwise may push into mania.