This is quite a philosophical, soft question which can be moved if necessary.
So, basically I started my PhD 9 months ago and have thrown myself into learning more mathematics and found this an enjoyable and rewarding experience. However, I have come to realise how much further I still have to go to reach a point where I could even think about publishing original contributions in the literature given how intensively everything has already been studied and the discoveries already made.
For example, I have just finished a 600 page textbook on graduate level mathematics. Although it took me a while to understand everything in it, I learned from this and enjoyed doing the exercises, but realised by the end that I still basically know nothing and that it is really intended as a springboard to slightly more advanced texts. I picked up another book which starts to delve more into one of the specific aspects in the book and again, it is 500 pages long.
Do I have to read another 500 page book to get a sense of something more specific which I can contribute? At this rate, it will be years and years before I am ever able to publish anything.
Later: I am reading this a few years later and realise the question could be hard to answer, as depends on many things (there are some problems where one could contribute decisively without knowing any math at all). However, I will leave the question as I think it's something that many students ask themselves and there is some useful generic advice in the answers.
Best Answer
It is something of a myth that everything has already been studied and that you have to master thousands of pages of prior work before you can contribute something new.
To be sure, there are some subfields of mathematics that are highly technical, and you're unlikely to be able to contribute something new to them unless you've studied a lot of background material. However, there are also areas of mathematics that don't require that much background knowledge. For example, Aubrey de Grey recently made spectacular progress on a longstanding open problem in combinatorics, and almost no background knowledge was needed for that problem. Even in supposedly highly technical areas of mathematics, people sometimes come up with breakthroughs that employ very little advanced machinery.
As others have mentioned, more crucial than "knowing everything" are (1) finding a good problem to work on, and (2) having problem-solving ability. If you have both of these, then you can typically learn what you need as you go along. When you're at an early stage in your career, finding a good problem generally requires an advisor, unless you have the rare ability to smell out good problems yourself just by reading the literature and listening to talks. Problem-solving ability is probably innate to some extent, but a lot of it comes down to experience and persistence. Of course you will be a more powerful problem solver if you have a lot of tools in your toolbox, but generally speaking, you get better at solving problems by spending your time directly attempting to solve problems, and only reading the 500-page books when it becomes clear that they are needed to solve the problem you have in mind.