Can I be covered under other insurance and still be eligible for an HSA?

If you are covered under anyone else's non-HSA eligible High Deductible Health Plan, then the answer is no¹. However there are certain coverages that are considered permitted by the IRS (you will not be disqualified from opening an HSA account with the following):

1. Insurance coverage for something that is non-medical (e.g., workers' comp, home insurance, auto insurance, etc.)
2. Insurance coverage for specific disease or illness (e.g., cancer policy)
3. Hospitalization insurance where insurance pays a fixed amount per time period (e.g., days or weeks)
4. Coverage under an Employee Assistance Program
5. Discount cards to health services
6. Insurance coverage for dental, vision, long-term care, accident, or disability

Source: IRC Sec. 223(c)(1)(A)(ii)