Can one customer belong to two customer groups at the same time?

No. A customer has one Customer Group at a time. On their Profile tab the field is a single drop-down, so choosing a new group replaces the old one.

If someone needs both sets of benefits, create a separate group for them, for example "Wholesale VIP":

1. Go to Customers → Customer Groups and click + Add group.
2. Add the discount rules and Free shipping for this group if needed.
3. Open the customer under Customers → Manage Customers, pick the new group and click Save changes.