Ehhh, that’s blanket advice that might not necessarily apply. If you have a few grand in an emergency fund, it’s not awful to start putting money in a Roth IRA or something you can withdraw the contributions from. Yeah, maybe if you get unlucky you’ll take a loss if you have to sell to supplement your not-quite-adequate emergency fund. But if you don’t, and you don’t have any emergencies (and you only use your emergency fund for emergencies), then I don’t think it’s too horribly risky to be investing before you’ve got 3-6 months, as long as you have enough to cover 1 or 2 large unexpected bills.
For example, my expenses are about $2500 a month. My emergency fund is about $6500 right now, so that’s like 2.5 months, less than the recommended 3-6. I’m saving $500 a month toward my emergency fund to bring it up to par, but I’m also putting $1500 a month in a Roth 401k because that money is going to grow so much in the 30-40 years until I retire.
If I followed your advice, I’d miss out on Roth contributions for several months (which would also be a horrible waste of the favorable tax situation I’m in this year that makes Roth such a good idea) while I build up a pile of cash that will do nothing but sit there and lose value to inflation, for a 6 month long emergency I’m really not very likely to have. My $6500 will go a long, long way toward fixing any problems that might arise, and the $500 more a month I’m throwing at it will only help even more.
Plus, if in some vanishingly unlikely twist, I had an emergency that depleted all $6500 at once, I could immediately stop the Roth contributions, put expenses on my credit cards temporarily, and just wait until the very last minute to pay them (if I time expenses right, I can float them for almost 2 months). The float would unlock another $3k of money that would have been going into the Roth that I can now use toward the emergency, as long as the emergency isn’t me losing my job.
TL;DR: blanket advice isn’t always great. It’s a matter of weighing the risks you think you have against the possible great benefits of investing some money while continuing to bring your emergency fund up to snuff.