I don't mean to categorize people in what I write below. It is just a personal opinion and an attempt to lay out different ways of becoming a bioinformatician that are rooted in our individual preferences.
In simple terms, one could argue that becoming a bioinformatician is simply a matter of adding some biology to an existing IT/programming/informatics knowledge or vice versa. One can do a whole RNAseq experiment, from the wet lab experiments to data analysis, with limited biology knowledge other than RNA expression, and with just enough ability to run prepackaged scripts and make tiny modifications to them such as file locations or p-value cutoffs. I have interacted with them frequently, and they do just fine for themselves even with the aforementioned limitations. I wouldn't want to train students with such limited outlook on research, but there is not doubt that ultra-specialization works.
A bigger question for me is why anyone wants to become a bioinformatician, and from there it becomes easier to decide what the missing ingredients are. Some people want to get into bioinformatics because it is hot and pays well - or at least they think so. As I said above, there is nothing wrong with earning a living by knowing just enough of biology and these other fields. In fact, some people who are in this group are so good that they become irreplaceable.
Then there is a bioinformatician who is an excellent collaborator, and is stronger on one side than the other. These are in general very capable and professional, and can get away with being much stronger in biology or in everything else because their expertise is complementary to the whole team rather than driving the whole project.
The most difficult to become, and thus rarest, is a bioinformatician who can lead a group and drive the whole research endeavor. To achieve this means spending a lot of time learning about science and its process in general, and not just biology or just computers/programing/math. It means being able to talk with people who know more in their area of expertise but not necessarily can synthesize all the information. It involves life-long learning to stay abreast of developments in all areas. Not very many people would have this as their goal early in their careers because hardly anyone decides when they are younger than 30 that life-long learning will be what they do. It is usually something that develops later, so I think this group for practical purposes can be eliminated from the early consideration of what kind of bioinformatician one wants to be.