As with all things there is no black and white answer. It always depends.
Most of all it depends on where you work and what exactly your job entails.
If you are a freelancer or in a small web team then being multidisciplinary is crucial. After all there will be few people to fill gaps in your knowledge.
However, if you work in an agency or as part of a large web team, the chances are you are one of many specialists and the need to be multidisciplinary is less.
We should all cultivate a broad understanding
That said, I think it is beneficial for us all to have a wide (if superficial knowledge) in a lot of areas. This enables us to understand and appreciate other peoples specialisms.
For example if I understand how to write PHP, even if at the most basic level, it helps me in my dealings with somebody who specialises in that.
This improves working relationships, enables projects to run more smoothly and leads to better sites.
That is why designers should know how to code
That is why I believe all designers should be able to code. Even if they are not required to do it as part of their job, they should at least have an understanding of the process.
Think of your skills as a capital T. You should have a broad superficial level of a lot of things and then a deep understanding in one area.