You also need to take into account the size of each response, how long your server needs to keep the data in memory (e.g. because of latency, the requester's bandwidth, etc), whether requests to the same file can share a buffer or not, how much data you can be sending at the same time while still being responsive (e.g. without slowing down other responses, causing them to take longer, requiring you to keep those resources in memory for longer, and snowballing from there), ..., stuff like that.
For short text messages, probably not an issue. With larger stuff like images or video, I would be more careful.
Still, even for text-only, if you're using PostgreSQL, by default you have a limit of (I think) 100 parallel connections (or 97, because I think 3 are reserved for superusers), but each connection can only be executing one transaction at a time, so that can quickly become a bottleneck depending on your application and how fast you need to make queries vs how long your queries take to return a response. So then you might need to tune some PostgreSQL settings, or add caching, or some other way to work around the issue.
If you add more services, then you also need to keep in mind the latency between those services.
And so on and so on. So RAM and network would probably become an issue way earlier than CPU in most cases.
For short text messages, probably not an issue. With larger stuff like images or video, I would be more careful.
Still, even for text-only, if you're using PostgreSQL, by default you have a limit of (I think) 100 parallel connections (or 97, because I think 3 are reserved for superusers), but each connection can only be executing one transaction at a time, so that can quickly become a bottleneck depending on your application and how fast you need to make queries vs how long your queries take to return a response. So then you might need to tune some PostgreSQL settings, or add caching, or some other way to work around the issue.
If you add more services, then you also need to keep in mind the latency between those services.
And so on and so on. So RAM and network would probably become an issue way earlier than CPU in most cases.
TL;DR: "It depends".