WASHINGTON — For working people making modest wages and struggling with high medical bills from chronic disease, President Barack Obama’s health care plan sounds like long-awaited relief. But the promise could go unfulfilled.

It’s true that patients with cancer and difficult conditions such as multiple sclerosis or Crohn’s disease will be able to get insurance and financial help with monthly premiums.

But their annual out-of-pocket costs could still be so high they’ll have trouble staying out of debt.

You couldn’t call them uninsured any longer. You might say they’re “underinsured.”

These gaps “need to be addressed in order to fulfill the intention of the Affordable Care Act,” said Brian Rosen, a senior vice president of the Leukemia & Lymphoma Society. “There are certainly challenges for cancer patients.”

“Cost may still be an issue for those in need of the most care,” said Steven Weiss, spokesman for the American Cancer Society Cancer Action Network. That “makes it critically important for patients looking at premiums to also consider out-of-pocket costs when choosing a plan.”

Out-of-pocket costs include a health plan’s annual deductible, copayments and cost-sharing.

A few numbers tell the story. Take someone under 65 with no access to health insurance on the job and making $24,000 a year — about what many service jobs pay.

Under the health care law, that person’s premiums would be capped below 7 percent of his income, about $130 a month. A stretch on a tight budget, yet doable.

But if he gets really sick or has an accident, his out-of-pocket expenses could go as high as $5,200 a year in a worst-case scenario. That’s even with additional financial subsidies that the law provides people with modest incomes and high out-of-pocket costs.

The $5,200 would be more than 20 percent of the person’s income, well above a common threshold for being underinsured.

“Chronically ill people are likely to be underinsured and face extremely high out-of-pocket costs,” said Caroline Pearson, who tracks the health care overhaul for Avalere Health, a market research and consulting firm. “While the subsidies help, there still may be access problems for some populations.”

Obama administration spokeswoman Joanne Peters said the new system is still “night and day” from what patients faced for years, because insurers can no longer turn away those with pre-existing medical conditions, and because the new plans cap out-of-pocket costs.