Earlier quoted context omitted.
You can compile Bootstrap yourself from LESS or SASS, pushing your own values into the Bootstrap variables (and ideally sharing this configuration with your custom styles). In the process you can choose not to include some Bootstrap components, reducing it's weight significantly. It's NOT a lightweight framework.
I know I can do this, but I don't because I prefer to just download the boostrap latest vanilla, and override as needed. That way every time there is a new version I don't have to do mental gymnastics on what variables are new or renamed.
At least the SASS-port of Bootstrap declares it's variables as defaults - meaning they would not be set if they were set earlier (in your configuration). This makes upgrading the gem (or plain source files) rather simple. But sure, you might have to check if anything was renamed or added.
But with overriding you do not only create redundant, unnecessary styles (and bytes), but if you now upgrade your bootstrap files as you suggest, you also have to change your overrides - which depending on the amount of changes you did will be significantly more work (and 'mental gymnastics') - if changes were introduced.
So you gain very little (less setup effort) but lose quite a lot.