The other answers already do a good job at explaining why the limit cannot be raised. This answer will try and explain what the limit is and why it exists in the first place.
The reason why many keyboard won't allow more than 2 keys at the same time (with a few notable exceptions like Ctrl + Alt + Del), is because of a feature called "anti-ghosting".
What is anti-ghosting and why do I need it?
As the name implies, anti-ghosting is a feature meant to prevent ghosting, which is a bug that appears as soon as people attempt to press more than 2 keys simultaneously. As soon as the 3rd key is pressed, the keyboard will see a 4th (ghost) key being pressed.
How is ghosting even possible?
In the past, each key on a keyboard was attached to its own unique wire, so the keyboard knew exactly which keys were pressed. In order to reduce costs, manufacturers chose to reduce the amount of wires needed to map all keys. To do this, they created a grid and mapped the keys to the grid.
Consider the following keyboard: It's a simple keyboard with 9 keys (numbered 1 to 9) mapped on a grid, which uses 6 wires (instead of 9), 3 horizontal ones, and 3 vertical ones.
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If I press key 7, the left column and top row will send a signal to the keyboard. The keyboard knows which key I pressed, because there is only 1 key mapped to both the left column and the top row.
If I press 8, the middle column will also fire, but since I haven't let go of 7, the top row is still lit up. At this point, the keyboard knows that I'm pressing 7 and 8 simultaneously, because those are the only possible keys that can light up both the left and middle columns, which are both mapped to the top row.
But what happens if I press 5 as well? At this point, the middle row will light up, but the keyboard has no idea which column the signal belongs to, since the left and middle columns are both being pressed right now. Because of this, the keyboard will assume that I not only pressed 5, but also 4 as well.
Key 4 is a "ghost key", because it appears as being pressed even though it's not.
But I can press Ctrl + Alt + Del, no?
Yes. Depending on the mapping of the keys, it is possible for the keyboard to detect 3 or more keys simultaneously. Most manufacturers design their keyboard in such a way, that people can press multiple modifier keys (Shift, Ctrl, Alt) together with at least 1 additional regular key.
However, while this is good enough for most users, it's not enough for a gamer that needs to press W + A + Ctrl + all at once, and another gamer might need a completely different combination of keys altogether.
So how do I know what keyboard I should buy?
Luckily, there's a term for it: "rollover".
While different keyboards may support different combinations of 3 or more keys, all of them are guaranteed to recognize at least 2 keys simultaneously. This is known as a 2-rollover, which means any 2 keys will be recognized.
Many gaming keyboards mention their rollover. Some may be advertised as 3-rollover (any 3 keys work) or 5-rollover, some high-end keyboards may even be 10-rollover (any combination of 10 keys are recognized). The most expensive keyboards may also be advertised as "n-rollover", which means that you can (in theory) press all keys on the keyboard at the same time, and they would all work.
The term may vary depending on the manufacturer or the store that sells them. For instance, I found a keyboard on Amazon that claims to have "19 Non-Conflict Keys", which likely refers to a 19-rollover.
Naturally, the higher the rollover, the more expensive the keyboard.
As a general rule-of-thumb, if the keyboard is not aimed at gamers, chances are it's only 2-rollover and therefore suffers from the same 2-key restriction. Be careful, though! Not all gaming keyboards are guaranteed to be 3-rollover or higher; some may simply be optimized for WASD despite being only 2-rollover. If you want to avoid any surprises, pay attention to the keyboard's rollover.