What it actually means: the product doesn't intentionally contain nuts, but it's made in a facility that also handles them. There's a real risk of cross-contamination during manufacturing.
It's a legal protection. Manufacturers use these warnings to protect themselves, but it doesn't mean the product is definitely unsafe — only that they can't guarantee it.
Your call. Many people with severe allergies avoid all "may contain" products. Others contact manufacturers about cleaning procedures. There's no one-size-fits-all answer — talk to your allergist.