Stop nagging and I'll get you one. This is the third new purse she wants.
I tried "I need a new case," thinking of a guitar case, or a glasses case. Would I have to be more specific?
Only if you're using US English. In some countries (e.g. Australia) a woman's 'purse' as used in the US is never 'purse' but 'bag,' 'shoulder bag', or 'handbag'. A 'purse' in Australia is a woman's wallet. We would never call the bag a woman is holding/wearing a 'purse'. An Australian would never translate ארנק as 'purse' unless they were on Duolingo and knew they would get marked wrong if they called it a 'bag'. :-)
Does it? It may be based on US English, but it makes a point of accepting many other variations: 'colour' as well as 'colour', for instance, and other non-American spellings, but also non-American ways of saying things: 'shop' instead of 'store' and lots of other examples I can't think of right now. It seems reasonable to me.
And what would we call the guy that collects the money on airplanes and ships, you know, the "purser"? Turns out, it's a completely different word...גזבר, meaning more like "treasurer".