Is it possible for a bipolar person who's on meds to still have an episode?

Yes. There are many possible reasons why a person with bipolar disorder could have symptoms, even while staying consistent in treatment. Medications are often helpful, but even under the best of circumstances people may have "breakthough" symptoms. This is one reason why it is important to stay in regular contact with a physician, so that quick adjustments can be made in medications if needed.
Yes. There many possible explanations for relapse in bipolar patients (bpd). Alcohol, drugs, tobacco, caffeine, illnesses, insomnia, psychosocial stressors, and other medications used intercurrently that interact with maintenance medications can throw bpd into imbalance.
In the best . possible scenario - a person's up's and down's are kept within a range that is not too wide. However, as stated earlier - there can be breakthrough cycling even with appropriate medication and healthy lifestyle. In that situation the symptoms are usually not as pronounced as they might have been off of medication. Talking therapies are also very important. .
Alas yes. A relapse can occur even when a person is engaged in treatment; even the best treatments help the person to control their illness but generally do not eliminate it ("cure"). If an episode breaks through, it is important to involve one's physician for adjustments to the treatment plan (different dose, different med, different kind of psychotherapy, etc).