Competition is good. Competition is needed. Yes, you can have competition and collaboration at the same time. That is what separates the okay teams from the great ones.
I have been an athlete my entire life. In my experience, competition is what pushes athletes to do better and be better. That constant push to get better takes a drive that is inside every competitive person.
Having said that, my entire athletic life has been on teams. I have never competed in anything on an individual basis. So, to me, working together (collaboration) is a necessity for teams to achieve their goals. It is about sacrificing individual stats for the betterment of the team.
When does it go awry? When people are more focused on themselves over the team. Think about teams that you have heard described as having "that it factor". To me, that factor is a collective understanding that the success of everyone is far more important than the individual. As the famous saying goes: "The name on the front of the jersey is more important than the name on the back of it."
Collaboration helps people push towards a common goal while sharing resources and talents. A great team is a tremendous example of this.
Competition is good. Competition is needed, but it must be the kind that keeps the focus on improving the team first, the individual second.