Can a furniture company be held liable if an injury was incurred due to shabby assembly of their employees?

From what you described it does sound like it could have been negligent assembly and set up of the bed because beds should not ordinarily collapse when they are being moved just a little bit.
You should contact a local personal injury lawyer to investigate, take photos, measurements, and help prove the case.
The two elements are liability or your proof that the furniture company was at fault and damages or your proof of your injuries.
Our firm has handled similar cases in the past including one where a client rented furniture and the delivery guys knocked something over while carrying a couch causing an injury to our client. Im sure there are other precedents in the case law where plaintiff’s recovered in similar situations.