There is a very funny comic at phdcomics.com that shows that as one progresses from undergraduate studies to Phd, one changes from knowing a little about everything to knowing a lot about something. What you don’t want is to know everything about nothing!
The comic does make one conscious of the need to have broad knowledge. It is good to have an idea of the big picture and know how one’s field stands in connection with others. I have always felt that the US PhD programs with their initial coursework ensure a broad knowledge base. If one is doing a PhD from the UK, it is better to have an MSc under your belt. More importantly, one has to make a personal effort to talk to people from different backgrounds, read widely and attend departmental seminars.
I was recently talking to Imre Leader and he was telling me about a book he is editing which aims to familiarize mathematicians with fields that are not even close to their own area of focus. The book is the The Princeton Companion to Mathematics. The chief editor of the companion is Timothy Gowers who is a Fields Medal winner. The Fields Medal is considered as being equivalent to a Nobel Prize in Mathematics. Anyway, the companion is a commendable effort. It is written in an extremely accessible fashion and a lay-person can benefit from it. It might help us know a little bit about many things, even if in this case it is only within the field of mathematics!