Yes, it was already in place when I was hired. I'm a Drupal developer, and pretty much every site that I work on uses 10s or 100s of contributed modules from Drupal.org. When a bug is reported to us, the first thing we do is narrow down what module is causing it and then go look through the issue queue for that module to see if it's already been reported. If so, then we can just apply the patch from that issue. If not, then we fix it and upload the patch to Drupal.org. We're already doing 99% of the work involved in open source contribution. Why not do the remaining 1% and get the benefits from it?
There's a clear benefit to doing this. If we have a module that we have to apply 100 patches to in a specific order every time we upgrade the module, that's a lot of maintenance burden. If we submit those patches upstream and let the module maintainer roll them into the module, then maybe we only have 4-5 patches to apply. That makes upgrades significantly easier. In fact, our developers are the maintainers of some of the modules that we use, so they have incentive to be responsive maintainers. When a patch is posted, it's in their best interest to commit it ASAP so that we don't have to include that patch in our patch list.
Getting to that point is step 1, and it's relatively easy if you work in an environment where you depend on a lot of external open source software.
The next step is slowly pushing a cultural shift in your company to develop things in a generic way that would be useful to people outside of your organization. Unfortunately, there's no clear cut way to do this, as it depends on too many factors. Again, though, the justification really comes down to maintenance burden. If it's open source and people use it, then it's likely that other people will help you maintain it (fixing bugs, adding features, etc). The more you involve the community, the more likely it is that you will be able to share the maintenance burden. An example of this is that a few of our developers co-maintain this module: https://www.drupal.org/project/media_theplatform_mpx
We needed to integrate with a video service, and since it's a third party commercial service, it's likely that somebody else needs to integrate with it as well, so we did all of the generic integration in the module I linked to before, and then the very few things specific to my team's product go into a module that we don't open source.
One thing that I've noticed is that it really comes down to your supervisor being open to looking at things a little differently, but I think the long term benefits are clear and definitely worthwhile.