So there are a number of things to consider here.
Hiring takes longer than people think
Companies underestimating how long it will take to get back to a job candidate is far from rare. It's arguably more common than the opposite. A lot of people are involved in a hiring process and they are very often quite busy. Especially for decision makers recruitment will be an added responsibility on top of their workload and their response time is something internal recruiters can't control. Some recruiters will therefore pad their timeline based on past experience. Some will give you the time that they hope to get back to you, realistic or not. Plenty don't bother giving a time line and there are any number of reasons for delays that could push whatever the original time line was.
You specifically mention a "time limit" and "maximum allotted time" in your question but that's really not how this works. A hiring company is in control of the timeline and they will typically tend towards optimistic estimates. Right now a hiring timeline is especially hard to predict because:
We're in the midst of a pandemic and an economic downturn
Many companies are suspending their hiring process while almost all are reducing the number of new hires or introducing new approval processes. That means that what used to take a few weeks can now easily take a month. The kind of delays you describe are very much par for the course in any economic downturn and that's before you consider the impact that going remote can have on antiquated administrative processes. A simple employment contract that used to be a signed hardcopy might now cause a real headache for recruitment teams.
Good recruiters will acknowledge delays
But there are plenty of bad and average recruiters out there. Ideally the person you spoke to would have acknowledge the delay and perhaps given a very good (pandemic-related) reason for it. Certainly if it's been several weeks they should update you on their new timeline both as a courtesy and to avoid losing you as a candidate.
When they don't acknowledge it at all, that's when I would push on the subject, ideally in a call or during the next interview. Their response will tell you much more than guessing at the reasons. ALso keep in mind that one unprofessional recruiter is not in and of itself a deal breaker because:
Delays in the hiring process might not impact you on the job
When you're talking about red flags in a hiring process, you're talking about things that could negatively affect you in the job you're applying for. Unless you're applying for an HR position it's highly unlikely that you'll have much to do with the people in charge of recruitment. It could signal some level of administrative incompetence within HR in general meaning problems and delays in payroll and holiday approvals for instance. But even that isn't certain and it might not be something you'd consider an outright red flag.
Generally speaking, slow processes and long delays in hiring will tend to map to work cultures that are also heavy on red tape. But it's far from universal.
What does that mean for you?
Overall, I'd take this as any other data point about a potential employer. Their process is slow but that's understandable. It sounds like they didn't acknowledge the delay which is somewhat unprofessional but again: pandemic. If you see other signs of severe administrative hurdles or other red tape, you can assume that it might be a thing for this company in general. If that's a potential issue for you I would definitely ask the hiring manager (the person you'd work for) about it.
Personally, I accepted a job with a company where the hiring process was concluded within 5 weeks, but the actual offer took another 4 months to materialise as a result of the pandemic. It was a clear sign to me that this company is process-heavy and likes its red tape. The experience on the job confirmed this! But the most that this impacts me is that tangential things like bid management and time sheets are more of a chore than in my previous company. Because I knew that, these delays ultimately were not a factor in my decision to accept the offer when it did come. (Though I should also mention that almost all firms suspended or delayed hiring during this team. Hopefully a disruptive event on this scale won't happen again any time soon.)