Can you have insomnia and hypersomnia at the same time?

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.
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.