Getting SSI does depend on being very poor. If you're getting SSI, they do review your financial status on a regular basis, at least yearly. A sizable inheritance certainly could get a person kicked off SSI.

SSDI, on the other hand, is granted to a disabled person who has worked long enough to qualify. A billionaire can get SSDI just as readily as a poor person. Your eligibility for SSDI has absolutely nothing to do with your financial status. Once you get SSDI, you will never lose it due to inheriting a lot of money. The same is true for Medicare. (You can only lose SSDI by regaining your health to where you could return to work, and the SSA discovers that.) Qualifying for Medicare has nothing to do with how much or how little money you have. Medicaid, on the other hand, is only for the poor.