Can you have insomnia and hypersomnia at the same time?

Yes!: I have had many patients who reported having trouble falling asleep during the night, but also stated they slept a lot during the day. This type of insomnia/ hypersomnia combo is circadian in nature and typically common among sedentary patients with little sun exposure, and those with excesive caffeine use, anxiety, or irregular sleep-wake schedules.

No: Obviously you can't fall asleep and not fall asleep at the same time. But people with sleep disorders might experience these different symptoms at different times.