For me, this is a perception versus reality discussion. Coworkers generally assume I'm a SQL Server specialist, maybe even they think I'm more specialized than that and I only care about SSIS. Au vibrator (I'm blogging this on my phone and swype does not recognize contraire and corrects to "vibrator." It's funny, laugh), what I love is problem solving and I am pretty good at using a computer to solve problems.
In college, Professor Walters said if you don't like reading and aren't interested in life long learning, you were in the wrong field. He was right. I graduated with a degree in computer science and the ability to write the most amazing c++ code known to man. I actually did think I was good at it but looking back, I'd have had lunch handed to me had I ever interviewed for a c++ position. Instead, I landed a job working on an erp system with a c-like language. What a horrible, misguided year and a half that was, but I learned some database stuff (ingres) and a 4gl for that position.
Hating what I did provided the impetus to take a course on java. That lead to creating a monitoring app (production java experience) and I lucked into maintaining a new asp site. I wrote html by hand and vb was not hard to figure out so I left that job with plenty of things I wasn't planning on having experience with (I've had a recruiter ask about the postscript experience I pickler picked up there. Stack based languages are cool but omfg what a mind scrambler).
Since then, I've learned so much. Some of it is applicable daily while other bits will probably never be ever again useful. For the past few years, I've really focused on sql server for three reasons: I enjoy it, the pay is good and you, my #sqlfamily. More than anything else, it's the later that keeps me here. But, not to the detriment of acquiring a wide breadth of knowledge. Presenting at the user group and SQL Saturdays has really opened up a new avenue of learning for me, all those soft skills! That's going to apply well beyond when SQL Server is nothing but a fond memory. Making reports is easy but making good reports, that's hard. There's a whole field of information design and data visualization that I need to learn to be able to effectively deliver information. I may not like making GUIs but a good report is a little slice of heaven.
The pattern of learning whatever it takes to solve a problem is invaluable to remaining relevant in today's job market. Specialize all you want, just make sure the focus is on the process of learning and not the technology, because that's going to change.