Yes. Stock plans often have various provisions on any acceleration of vesting, on whether unvested options or restricted stock continue vesting, and on post-termination exercise periods for vested options. These can depend on whether the termination is "for cause" or involuntary, or whether it results from death, disability, retirement, or a change in control. Most commonly, when you terminate employment under normal conditions, vesting stops. See your stock plan and grant agreement for specifics, such as those on disability, death, and retirement.
Alert: Your company is not required to give you any notice about these provisions beyond the terms stated in the stock plan. Understanding the terms of your stock plan is your responsibility. The issue can influence your decisionmaking about how and when you leave your company.
See other FAQs for survey data on the most common treatments of unvested stock options and restricted stock/RSUs at termination.