Yes, it's both possible and legal for your health insurance company to drop you under certain circumstances. Being aware of when and why this can happen is important for your financial well being and peace of mind.
The good news is that health insurance companies in the US are regulated by various laws on both state and federal levels. This means that technically they can only drop members who meet (or don't meet) specific criteria.
The bad news, of course, is that sometimes those laws are confusing and their application to real-life situations involves a lot of fine print which not all customers understand. Furthermore, while you can not be dropped for just any reason, you can be dropped at any time the company deems proper.
Specific details of your plan will be outlined in the paperwork provided by your employer and/or your insurance company, so familiarizing yourself with that information is your most important safeguard against being unexpectedly dropped from your plan.
With that said, the reasons for which an insurance company can drop a member fall into a few general categories, and being aware of those is a good first step. In general, then, your health insurance company can drop you if:
You commit fraud.
This is kind of a no-brainer. If you misuse your insurance coverage in any way, you're breaking the rules of the contract, and the company is under no obligation to continue providing their services.
You lose your job or your situation at work changes in certain ways.
Many people receive their health insurance through their place of employment. If this describes you, then be aware that your job is a criteria for you being eligible for the coverage. If you leave your job, you also forfeit the coverage. In most cases, employers only provide health coverage for full-time employees, so if your full-time status changes to part-time or contract, your insurance will also end.
You don't pay your premiums.
If your insurance is arranged through your job, then you've likely noticed that a premium for it is deducted from your paycheck. In cases where this isn't done automatically, you run the risk of being dropped if you fall behind on premium payments. Some places offer a grace period for unpaid premiums. Many do not.
It's a common misconception that your health insurance company can drop you if you become seriously ill. As of 2010, the Affordable Care Act made this practice illegal for all policies written after the ACA-effective date of September 2010. If your policy was written before this date, you should look into its specific policies ASAP to be aware of where you stand.