Manage Your Skills, Manage Your Career
People change careers all of the time. Skills often get out of sync with personal goals. This can be an especially acute issue for project managers.
Many PMs simply like managing projects. To this group, moving from project to project is a great career. They may end up doing that forever! It may or may not involve switching among organizations and industries, but also many involve maintaining stability within one organization for a long time. It all depends on the person.
The key for project managers, however, is to try to find a balance between what you like, what you're best at, your goals, and the evolving needs within organizations - your current one or ones where you might like to work. You may wish to enter a career operational position within the company, for example, to more closely align your skills with the career ladder within the company.
The way to the executive ranks is not necessary up the project management ladder. It could, however, involve stints in project management at one or more points, but also moving into other areas. It is good for PMs to possess non-PM experience, just as it's good for non-PMs in their current jobs to possess PM experience and expertise!
So, the million dollar question for you is, "Is project management an end in and of itself for you, or is it a means to and end?"
PMs naturally often operate "outside" the system. Initiatives of an operational nature are kept within the organization, but when change is needed, a project is created. For project managers, this can be both a blessing and a curse. Success managing projects has the potential to keep you managing projects - and thus isolating you from the operational roles that might really advance your career. Likewise, success in operations can keep you from moving into a career-broadening project management role.
There is not necessarily a "project management ladder" within companies. The nature of project management is that it is separate and distinct from the day to day operations. However, because projects touch on so many different aspects of the organization, it is a great place to be to get exposure and build skills that will be helpful later within the organization.
It is natural that an organization will be led by operations, finance, technical, marketing, sales, or service-oriented people based on the business of the organization. Some organizations are actually very project focused - consulting and construction, for example - and the best talent develops top skills in project management and the service which the organization offers. However, many organizations are marketing oriented (i.e. Proctor & Gamble), finance-oriented (i.e. banks), technical (engineering companies), etc. The good news is that project management is a common thread that goes through all organizations in some way, shape, or form.
One thing that organizations can do to better manage their talent is to develop a talent management system. Such a system is essentially a database within the company that monitors the various skill levels of the work force in a macro sense against evolving needs within the company. It's a succession management application or sorts, and most likely is led by the HR team. Just as a sports team has one or more backups for each position - in case of injury, retirement, or turnover - a company needs to have the same thing. Department and business unit manager surely do this to some degree, but a larger organization can provide the possibility of much greater career opportunity to people and a bigger bench to the organization.
As a project manager, you need to manage your own career to a large extent, but placing yourself in the right organization can make the job a lot easier. If an organization has a talent management system, that is helpful and is an indication that they treat the subject as a priority. If it does not, that could be a good project to suggest! In any case, you need to be mindful of the various opportunities and pitfalls around your career and manage accordingly. For an interesting PM-oriented take on this topic of talent management, see Mark Mullaly's article "What We Are Getting Wrong About Talent Management" at www.projectmanagement.com/articles/304456/What-We-re-Getting-Wrong-About-Talent-Management .