I think that for most people, if they abandon anything that's not already a huge success three months in, they'll just end up bouncing from one thing to the next and never seeing anything through. It's virtually impossible to bootstrap that quickly, and I'd be surprised if many of the big VC-backed startups we've heard of hit that either (certainly not Google or Facebook who weren't even monetizing yet at that point).
Of course there are scenarios where $10K MRR is possible quickly, but there's no way that is what most successful software businesses look like in 3 months.
It's really a way to frame the product launch by forcing you to ask hard questions about what you're doing. If you don't even see a way for the product to reach $10k/month with a short-term plan, maybe it's a side project rather than a startup? Or maybe the product niche is too vague or unprofitable, and you need to work on that first?
I'm still a crap entrepreneur, but I've become slightly better at teaming up with people who can make it happen.
If it's a side project, that's a different game. A side project doesn't need traction to be successful for its creator.
I don't know how much Atlassian made in its first three months, but they are one of the only bootstrapped tech companies to go public in the last 2 decades. It took them 15 years. While this is not necessarily the fastest path to success, I 100% believe it deserves immense respect. They're one of the biggest success stories in tech as far as I'm concerned.
(Of note, Atlassian has mainly focused on enterprise markets)
To answer your question: The day all your other investment dries up in the worst case. If you have no job or savings you can probably live 3 months on things like your credit cards and not paying the landlord - after those 3 months you better have enough income to start paying your personal debt. That is you need to send the landlord a check for at least 1.5 months rent and with a promise to pay the rest off soon. Likewise you need to be able to buy your groceries with cash and pay down the credit card dept you ran up. It should go without saying that in those 3 months your personal bills should be just enough to stay alive.
Note that the above is the worse case. If you are in month two of the above situation and realize that you won't make the $10kMMR number you should cut your losses, declare bankruptcy and find a day job.
You probably have some savings to live off of though. This means that you have some flexibility. If you are in month 2 and realize you won't make the 10kMMM you can decide if you cut your losses now or not. You have an idea of what your overhead is, who your real competition is and what your leads are (and thus likely future MMR - though don't get over optimistic), if you think things will get better continue, if not cut your losses: you can probably sell to a competitor for a small price and make everyone happy.
If you have investors who are any good at all they have considered your business plan. Sometimes they will insist on the 10kMMR or they will exit, other times they will agree to fund a plan that won't make money for months - but the plan covers that and you have enough income from the investors to live.
There are always other considerations to. 10kMMR is a good number, but you have to figure out how it applies. If your target market is snow plows you should expect your entire income to come during the 3 winter months, and 0 the rest of the year. Thus in winter your MMR needs to be at least 40kMMR, and you need a savings plan of some sort to get through summer.
Don't forget that you can sometimes moonlight. Maybe your ideal is only worth $2kMMR. If you can support this business working only 8 hours a month you keep (or find) a day job for most of your income. That extra 8 hours a month works out to $100/hour after overhead. This again changes the calculations.
You probably need to add how much you can spend on customers acquisition to your equation.
You need to get your company to 10k USD monthly product revenue within three months. If you can't, either the product, target market or team needs to be revised drastically.
It's hard advice to follow, but it will save you a lot of time because you can't wait months and years doing unessential tweaks to the product and marketing, hoping that sales miraculously grow.
It's also useful as a pricing yardstick early on. If you have very few customers, they need to be paying enough that you reach $10k almost immediately. If your product isn't worth that much, then you need to scale out. It's best to figure this out right from the start.
If it feels like you can't do $10k MRR in three months on your own, then you need to find a cofounder who can do it together with you... So it's a good way to calibrate cofounder expectations as well.