What I've noticed is just conversing empathetically with others a la How to Win Friends and Influence People and having technical discussion is enough for others to sing my praises, from my time as intern through middle-manager. Basically, be interested in others over yourself. For more on the nuance of "interested" I'd suggest the above book.
Bragging is making your achievements known. Bragging poorly is doing this in a way that's too obvious & grating to the listener. If you're accused of bragging, this means you are bragging poorly.
But superiors working with you, observing you in action, will come to a conclusion too.
The difference I've personally seen in getting my projects through is my relationship with stakeholders; e.g., if one executive isn't on-board the project is toast. I've taken to communicating personally with potential naysayers early to consult with them to get their feedback but mostly to get the first impression over with before presenting it to a larger audience.
In business unlike socially it's actually part of the job - or it should be if you want to achieve certain goals: bonus, promotion, peer recognition. Even avoiding layoffs could come down to how well you communicate the good work you're doing.
This does not come naturally to many of us in this sector, but once you are aware of the need for it, there are ways to promote yourself authentically and without being too much of a show-off.