It is possible to collect both kinds of disability benefit at once under certain circumstances.
You could get SSI, for example, if your SSDI payments are low enough.
Some people who qualify for SSDI also receive SSI assistance to cover a gap in benefits.
Social Security requires a five-month wait from the time you are officially declared to have a qualifying disability to the time when you receive benefits. When your financial resources are low during those five months, you could get SSI pay for that period, even when you’ve been approved for SSDI.