I've been at my company for a number of years. The team is small, and I was initially hired as a junior web designer/front-end developer, but throughout the years I've had to learn print design, animation, mobile design, interaction design, creation of iPad apps, Wordpress administration/customization, PHP, user interface design, information architecture, you get the picture.... the culmination of which has evolved into my current role as a User Experience Designer. I love to learn and have been fortunate enough to get paid to do it.
It has been a magical ride, but I am no Unicorn. I would consider myself to have a T-shaped skill set. I have skills in which I excel, and some that I am merely proficient in, and I have the wisdom to know which is which
I wouldn't change any of it. I've benefited from all of it. However, it can be a daunting position to be in. Technology changes at breakneck speed, and keeping up with - as an example - the latest libraries, frameworks, accessibility issues, standards, etc. takes a lot of time and energy. Look at this list of deliverables for UX alone. I get tired just looking at that.
I would never discourage anyone from learning as much as they can, but expectations need to realistic for both yourself and your team/manager/company.