At least part of the reason for the delay is to obfuscate who reported you, in an effort to prevent retaliation. While I think that's commendable, and I'm sure it does that job just fine, I do believe that it leads to a whole different problem. The lack of clear communication about the delay tends, I think, to lead to false retaliation, where players lay blame and retaliate against people who had nothing to do with the warning. It's unfortunate, and I would think it'd be easy to fix with better communication about how the pop-ups worked.
While the weight of reports is applied to determining whether a pop-up warning is issued, the validity of those reports is not. They aren't checked against what you've actually done, because they don't actually represent any form of punishment or sanction. They're just a warning, intended to help you catch yourself if you're going on tilt. Again, I feel that better communication about this would be a great help to Riot overall.
Altogether, this means that yes, you can receive this pop-up purely from false reports. It's unlikely (because it's fairly unlikely to receive a high number of false reports that have high weight to them in a short period of time), but it is possible. But it isn't a punishment. It doesn't mean that you're close to a punishment, or that you will be punished. It's just a heads-up that you've been getting reported a lot.
Actual punishments are reviewed to ensure validity of the reports. If you genuinely haven't done anything, you won't be punished, no matter how many people issue blatantly false reports against you.
Keep in mind, though, that when I say false reports, I do mean reports that were issued on games where you genuinely did nothing that could be considered toxic. I do not mean reports that were issued on games where you did something toxic, but not what the report said.
As an example of that last: A report for intentional feeding on a game where they flamed both teams while not giving up any kills is technically not a "false report" according to the system, because there was still toxicity present, even if it wasn't the form indicated by the report.