Because that's not how it works. You don't just blow your budget on servers if most of them aren't even going to be used, you plan it out based on what you estimate the loads will be. They obviously try to plan out ahead for those extreme peak loads too but sometimes demand is far greater than they've estimated. Other times, asshole kids decide to DDOS the network.
Why you'd bring this up as if it's a Sony exclusive problem, I have no idea. Microsoft's had the exact same problem plenty of times in the past. Hell, they just had outages a couple of days ago, didn't they? I guess they haven't got enough budget either, right?
Budget isn't just for servers, its for disaster recovery, staff, oncall engineers and engineering resource to improve services which are being attacked/greatly utilised.
You can plan and load test based on current concurrent users and then calculate how high your ceiling is based on how many users it takes to cause it to fall over. There's always a finite capacity. The ceiling for XBL will probably be larger, but its still easy for the attacker just to send more traffic
With DDoS it isn't just the amount of servers you have, its the management of your routers at a network level, the optimisation of the API's which are being attacked and how quick you are to respond.
The simple fact is, with DDoS, is that it's a lot easier to make your attack larger than it is scale your network to handle it. Even if you had more servers, all the attacker has to do is change a couple of input fields to make the attack larger.
Although, as an engineer, you can see what's being attacked, and try and craft your services in such a way so that request is blocked but normal service can be resumed. Although, overall its just a cat and mouse game.
Using last Christmas as an example, XBL was down for a couple of hours whereas PSN was down for a couple of days. Since PSN is growing by the minute, and the PS4 is performing excellently, you can guarantee that Sony will be pumping a lot of money back into their infrastructure.