I found an article today titled “Strapless” by Aaron Gustafson (read it here) with some arguments for not using Bootstrap in your project. I use Bootstrap all the time, and I feel it saves a lot of time when starting a new project. Here’s my opinions on his points:
1. Bootstrap doesn’t solve your problems. Design is problem solving. The design decisions made by the creators/maintainers of Bootstrap solve their problems, not yours. You may share some of those problems—a need for responsive layouts, for example—but not others. You need a system that is tailored to solve your problems and only you (and your team) know what those problems are. Have you ever tried on an article of clothing that’s “one size fits all”? How well did it fit your body type? Unless you are absolutely average in all respects, probably not all that well. Solve your problems with your own Bootstrap-esque pattern library.
I actually agree with this point, but Bootstrap is a starting point, not a full solution. I like it because I can get something working really quickly, and can make the structure of a page, then refine it and change the look and feel later.
2. Bootstrap offers more than you’ll need. Bootstrap contains a lot of components and design patterns. It was created to address a wide array of project needs
Okay yeah, it is a bit bloated. It’s crammed full of stuff that you might never need – but actually that’s something I really like about it! If you suddenly need to add a carousel, there’s one built in. If you suddenly need to add a responsive wrapper for a Youtube video, it’s there. If you suddenly need to add a modal, it’s there. Same for lots of other really standard stuff.
Plus as long as you’re using the Bootstrap SCSS files and not the final exported version, you can pick and choose which sections you want, and comment-out the ones you don’t need. Here’s the _bootstrap.scss file for one of my projects where I’ve just uncommented the components that aren’t needed. These won’t be compiled into the final css, so there’s no bloat. But I can add them back in really easy if I need them. 
If this isn’t how you include Bootstrap, you should (IMO anyway), and here’s an example project to get your started.
3. Differentiating yourself from you competition is harder. Bootstrap sites have a very common look to them.
Yeah but a Bootstrap site doesn’t have to look like a Bootstrap site! It’s a starting point, a framework. Sure if you keep the default theme, it will look like a Bootstrap site, but you can expand on that and add your own styles.
The major benefit I find is that it makes your markup standard, which means anyone who’s familiar with the Bootstrap classnames will save time when switching between projects and having to figure out how it all works!
Check out the rest of Aaron’s arguments here.