More than that on 9/11. Every news site, online forum any online resource for news you can think of was unaccessible for hours. TV turned out to be the only news source.
I was working at an ISP on 9/11. We were still online but none of our customers were getting anything (dialup modems) ... so people started calling our tech support number to read them the news. Or to get info about loved ones. That was a long day. I left a different person than I showed up as. Didn't we all though.
Telephony (landline and cell) was also down due to extreme volumes.
I don't know that in an event like that today we would do significantly better. Literally the entire country stopped for this to try and find info, because the event was just so extreme; the stoppage of all international flights and grounding them in Canada and Mexico with no notice was a side note, when that's literally never happened before. And, as with most major disasters of this type, the main issue was not physical damage but lack of electrical power to operate data centers and other infrastructure.
A good chunk of transatlantic traffic runs through New York. That IMO is probably a major thing that I don't know we can handle.
This reminds me of the distant past. The death of Michael Jackson resulted in (almost literally) the entire internet suffering an outage.