If a cheque is made payable to Bob or Alice, it is unambiguous. It cannot be negotiated by Bill or Mary. And it can be negotiated by either Bob or Alice. It is a valid cheque.
When I started working in a bank over 40 years ago, all cheques had an "or" printed on them. They were normally "pay Bob or order" which meant that they could be endorsed by Bob in favour of another person. Some were written "pay Bob or bearer" which meant that they could be negotiated by any holder.
The issue is not one of the validity of the cheque, but rather one of whether a bank will accept it. The majority of the rules relating to negotiation of cheques arise from banking practice, including the bank's interpretation of money laundering compliance. Some rules arise from precedent where judgements have been handed down in courts, frequently outside this jurisdiction, and a small number arise from legislation. There is no clear set of uniform rules which are practiced and accepted by all banks.
So the answer in my opinion is that a cheque payable to Bob or Alice is a valid cheque, but could easily fall foul of some internal rule of either the drawers bank or the payees bank. If the amount is small, the cheque is likely to pass unnoticed. If large (thousands) then it will receive greater attention.
I believe that a similar series of issues arise if the cheque is made payable to both and negotiated in the manner suggested by Slugbreath.
Mugsgame, I'd like to know the outcome.
Gulliver (Editor, ..... which is overdue for refresh/rewrite)