Despite the dictionary.com clear definitions, this can be a very grey area in the workplace.
Humble: adjective 1. not proud or arrogant; modest: to be humble although successful. 2. having a feeling of insignificance, inferiority, subservience, etc. 3. low in rank, importance, status, quality, etc.; (more here)
Arrogant: adjective 1. making claims or pretensions to superior importance or rights; overbearingly assuming; insolently proud: an arrogant public official. 2. characterized by or proceeding from arrogance: arrogant claims.
What one person sees as acting humble, another sees as an inability to take credit. And the reverse is true as well, what one sees as being arrogant, another may see as level with the person’s ability, output or results.
I find the office environment and your immediate supervisor are the best places to look for clues on how to act. All the while noting there is a clear line between boasting and being gracious. As long as you aren’t claiming an ability to move mountains, while you aren’t able to clean off the paperwork from your desk or taking the credit for someone else’s hard work, you are probably fine.
The question also comes down to a fairly simple one, would you rather be known around the office as talented but humble or arrogantly talented?