I don't like it when people invoke "don't reinvent the wheel". Yes, if you're reimplementing standard library functionality, then you could talk about reinventing the wheel.
But a lot of the time it really means "Let frameworks do everything for you." Don't reinvent the wheel, but feel free to reimplement subsets of the wheel that aren't all too tailored to your purpose. In fact, if you're a programmer, you probably do that all the time.
Particularly when it comes to web design, there is no specific "wheel" to speak of. I'm all for using Bootstrap... as a framework, not as a template you copy and paste.
I can make exceptions for local web applications (like router interface panels) and small, one-trick apps, though.