Is it possible to have one MBR partition and one GPT partition on the same drive?

Technically, yes, the MBR and the GPT are stored in different locations.

Practically all GPT disks have a special "protective" MBR. It's possible to have a "hybrid" disk by just writing regular partitions to the MBR instead of the "protective GPT" partition.

Usually though it can get really annoying and risky to always keep both tables in sync.

See the article Hybrid MBRs: The Good, the Bad, and the So Ugly You'll Tear Your Eyes Out by Rod Smith for the pretty-much-canonical description.

Technically: Yes

Practically: No, don't do it. It is unmanageable and unsupportable.