Unemployment insurance and Social Security disability are two separate programs, with different goals. In general, if you are receiving unemployment insurance benefits you are ready willing and able to work, and are not disabled. However, receiving unemployment compensation after your alleged onset date does not preclude a favorable finding for your Social Security disability claim.
The Social Security Administration recognizes that a claimant should not have to choose between unemployment compensation and Social Security disability. There is even a Memorandum from the Chief Judge on the topic.
It makes sense when you think about it. Able to work for unemployment purposes does not necessarily mean able to work full-time. Under Social Security's rules, a claimant is considered disabled if that person is unable to perform sustained work on a regular and coninuing basis (8 hours per day, 5 days per week, or an equivalent schedule – see Social Security Ruling 96-8p). Further, if you are over 50, you may still be able to work at an unskilled sedentary job, but could still be found disabled under Social Security's Medical-Vocational Guidlelines.
Under these circumstances, there is nothing inconsistent about receiving unemployment insurance benefits and applying for Social Security disability.
Nevertheless, some judges just don't like it. And the judge is entitled to consider receipt of unemployment benefits when deciding your disability claim. So it is important (as always) to know your judge. Also, you should have a good explanation ready for the hearing of the circumstances surrounding your receipt of unemployment benefits.
Check out the article by Colorado Springs Social Security lawyer Tomasz Stasiuk for another discussion of this same topic.