Can the mentally ill hold religious beliefs?

Just a discussion here,every time someone commits a crime in which God told them to kill, or they were possessed, or something like that, their reasoning gets discarded and they get labeled "mentally ill" which takes precedence over any beliefs they might have.

Which leads to my question: Can the mentally ill have religious beliefs? If they bad things they do are a result of their mental illness, then aren't the good things? Is: "well, she did it for god, but she's mentally ill, so doesn't count" a neutral statement whether it's a good or bad act? Can religions claim the good acts of the mentally ill, but disown them when it might make them look bad?