Why Can't I Cash A Cheque On My Mentally Ill Mothers Account?

I do not know in what jurisdiction you live.  But in most jurisdictions you have to have a "Power of Attorney" to access an account in anyone else's name.

In England, if someone is incapable of managing their own affairs, their attorneys are legally bound to register their attorney-ship with the courts and keep accounts so that the authorities can check up that the money is not being misappropriated by the attorneys.

I know most about England, as I am one of my mother's attorneys.  My mother and father both named my brother and I as attorneys, but this was not activated for a long time.  My father was capable of managing his own affairs until the day he died, so his was never activated.  When my mother felt in need of help with managing her affairs, we called on the solicitor and he activated our powers of attorney for her at her request.  By the time I returned to the USA she was in a nursing home and, though happy despite frail health, was not able to manage her own affairs in any way shape or form.  At this point my brother and I approached the solicitor again and he registered our powers of attorney with the authorities.

It is common for people to appoint attorneys while they are still capable of doing so, and, where this happens, the problems of signing on their bank accounts for purposes of transacting business on their behalf are greatly reduced.  

If your mother cannot transact her affairs at the moment and has not named you, or anybody else, as her attorney, then you should get legal advice as to how to proceed.  If you are named as her attorney and have the paperwork to prove it, then you need to go through the appropriate channels to get it activated according to the rules of your jurisdiction.

No bank will allow you to transact on someone else's account without power of attorney as, without it, they have no proof that you are authorized to do so, by either the owner of the account, or the authorities responsible for ensuring that an incapable person is not being defrauded.