The safest answer is no: if you are disabled, you are not currently able to work, which is a requirement for collecting unemployment benefits.
Although officially Social Security says that receiving unemployment benefits does not preclude disability benefits, the claims examiners and Administrative Law Judges tend to consider your filing for unemployment as an admission that you can work.
The safest answers aren't always the best. There is a wrinkle here worth knowing.
You are disabled under Social Security if your mental or physical disability prevents you from holding down full-time work of a type that is generally available in the national economy. What if you could work, but only where an employer would be willing to accommodate a whole lot of problems? For instance, you could work if you could find an employer who would let you miss a lot of work, take frequent breaks to lie down, or work at your own pace.
There may be such employers, but finding them is not going to be easy. So you may be capable of working, but suitable work is not available in significant numbers in the national economy, meaning you should qualify for Social Security disability.
If this is your cae, you can tell the unemployment office that you are able to work, but only in suitable jobs – and you are looking but so far there aren't any. This position is inherently risky, but I have had many clients who simply felt they had no choice and couldn't survive without unemployment benefits. Those clients have usually lost at a disability hearing.
My general advice is: this is a tough choice, but don't apply for Social Security Disability until you are within sight of your unemployment benefits running out. It usually will take about a year from application to hearing.
Most important: don't lie, don't omit information, and be up front with both Social Security and your state employment agency. The last thing you need is to get a Social Security overpayment you have to repay, or worse, face possible criminal charges for lying. To read more about this situation, see our article on Collecting Unemployment While Waiting for Social Security.