It is used when he/she disagrees your opinion. It is a euphemism. It includes the meaning "I don't think so, (but I could be wrong)..." 그건 아니지 -> "It's not" 않나 -> (making asking form) 싶은데 -> "I think.."

To be honest... I don't think makes sense in Korean... I could be wrong, so I'll confirm that here in a few minutes.

I think it means "it's not like that "