There's a classic article from The Atlantic in 1982 that explains the bullshit. The article is astonishingly relevant, nothing has changed in the 35 years since.
Yes, but only as a function of their scarcity. Diamonds aren't actually rare. They're only made rare by diamond companies hoarding them and releasing tiny numbers of stones in order to maintain a high price. You certainly wouldn't buy diamonds as an investment, just in case a diamond company dumped their stockpile on to the market and the price collapsed.
Not just that. If you also count diamonds manufactured in the laboratory, then diamonds are pretty common and cheap items. That's why diamond companies put a lot of effort into convincing people that only "natural" diamonds count as proofs of love.
I've never understood this argument. Aren't they worth what people are willing to pay?