Use it when something is about to fail or end very soon, not just facing problems. With all these missed deadlines, the project has had its chips.
Pro Tip 2/3
Casual Tone
It's informal and might be used among friends or in relaxed settings, not in formal writing or speech. After losing yet another client, I reckon we've had our chips.
Pro Tip 3/3
British Origin
The phrase is British slang, so it's more familiar to UK speakers than to other English speakers. He lost his job and flat in the same week; he's had his chips.