Imagine the servers like a building and the internet as a road network (this is overly simplified, but it ought to work). A player comes from his house, takes the car to the building.
- the roads might be clogged (especially when someone tells a different shady building to send millions of cars to go towards the building as well, DDoS)
- The doorway might be clogged (after a server restart everyone tries to get in)
- There's just not enough accomodation to serve everyone inside.
- The system used to serve players might also have a problem, causing everything to stop.
The first is not Riots responsibility, though they can get more connections to more different ISPs (well, depending on the countries internet infrastructure), this will probably require hardware changes and possibly even a physical relocation as well.
The second is already solved but one just can't reasonably get a doorway large enough to handle so many players at once, have some patience when the server is restarting.
The third is solved by adding servers, but eventually as the playerbase becomes larger, systems that worked just fine for small amounts of players stop working with these large numbers (think EU split into EUW and EUNE). A split like this takes time and causes players that used to be able to play together to be split up.
The fourth can be a result of a bug or hardware malfunction or overload. The solutions to all of these differ and bugs will happen, but just adding new servers won't help much without software and hardware to make servers take over the load of a crashed server (and there's a multitude of other concerns as well). Designing such soft- and hardware takes time.
