I run a bioinformatics software company, have been in the field for over a decade, and have worked in scientific computing even longer.
I'll start with a quick answer to the bubble question: there are already too many 'bioinformatics' grads but there are not enough bioinformatics professionals (and probably never will be). There are many bioinformatics Masters programs out there that spend two years exposing students to bioinformatics toolsets and give them cursory introductions to biology, computer science, and statistics. These students graduate with trade skills that have a short shelf life and lack the proper foundations to gain new skills. In that respect, there's a bubble, unfortunately.
If you're serious about getting into bioinformatics, there are a few good routes to take, all of which will provide you with a solid foundation to have a productive career.
The first thing to decide is what type of career you want. Three common career paths are Researcher, Analyst, and Engineer. The foundational fields for all are Biology, Computer Science (all inclusive through software engineering), and Statistics. Which career path you follow determines the mix...
Researchers have Ph.D.s and tend to pursue academic or government lab careers. Many research paths do lead to industry jobs, but these tend to morph into the analyst or engineer roles (much to the dismay of the researcher, usually). Bioinformatics researchers tend to have Ph.D.s in Biology, Computer Science, Physics, Math, or Statistics. Pursing a Ph.D. in any of these areas and focusing your research on biologically relavent problems is a good starting point for a research career. However, there are currently more Ph.D.s produced than research jobs available, so after years in school, many bioinformatics-oriented Ph.D.s tend to end up in Analysis or Engineering jobs. Your day job here is mostly grant writing and running a research lab.
Bioinformatics Analysts (not really a standard term, but a useful distinction) focus on analyzing data for scientists or performing their own analyses. A strong background in statistics is essential (and, unfortunately, often missing) for this role along with a good understanding of biology. Lab skills are not essential here, though familiarity with experimental protocols is. A good way to train for this career path is to get an undergraduate degree in Math, Stats, or Physics. This provides the math background required to excel as an analyst along with exposure to 'hard science'. Along the way, look for courses and research opportunities that involve bioinformatics or even double major in Biology. Basic software skills are also needed, as most of tools are Linux-based command line applications. Your day job here is working on teams to answer key questions from experiments.
Bioinformatics engineers/developers (again, not really a standard term, but bear with me) write the software tools used by analysts and researchers and may perform research themselves. A deep understanding of algorithms and data structures, software engineering, and high performance computing is required to really excel in this field, though good programming skills and a desire to learn the science are enough to get started. The best education for this path is a Computer Science degree with a focus on bioinformatics and scientific computing (many problems that are starting to emerge in bioinformatics have good solutions from other scientific disciplines). Again, aligning additional coursework and undergraduate research with biologists is key to building a foundation. A double major in Biology would be useful, too. To fully round this out, a Masters in Statistics would make a great candidate, as long as their side projects were all biology related. Your day job here is building the tools and infrastructure to make bioinformatics function.
All three career paths can be rewarding and appeal to different mindsets.
If you haven't followed the NPR series on gene sequencing over the last few weeks, it's definitely worth listening to. I also did a talk a few years back at TEDxAustin on the topic that makes the connection between big data and sequencing ( http://bit.ly/mueller-tedxaustin ). Affordable sequencing is changing biology dramatically. Going forward, it will be hard to practice some parts biology without sequencing and sequencing needs informatics to function.