Laws often tell us what problems a society faces, and the U.S. Equal Employment Opportunity Commission includes an entire section forbidding employment discrimination on the basis of having a foreign accent. The pertinent section can be found in the following online document: https://www.eeoc.gov/eeoc/publications/immigrants-facts.cfm
The government would not ban such discrimination if it were not a recognized and widespread problem. The EEOC document acknowledges that a foreign accent can prevent a person from being hired for a job that he or she is perfectly capable of doing, which is clearly a shocking limit on opportunity.
In a 2016 article by Andrew R. Timming on employment discrimination based on accent, we learn that
The results suggest that the managerial respondents actively discriminate in telephone-based job interviews against applicants speaking Chinese-, Mexican- and Indian-accented English, and all three are rated higher in non-customer-facing jobs than in customer-facing jobs.
If accent is a barrier to jobs during an initial phone interview, it is clear that this is a significant problem for non-native speakers.
In a December 30, 2018 article Forbes magazine, Dr. Pragya Agarwal discusses the issues of workplace discrimination based on accent. Dr. Agarwal examines the psychology of judging people based on how they speak. The article offers possible solutions to the problem that include acknowledging our biases and including more people in the workplace who have a variety of accents.
Given the amount of research into accent bias in the workplace we can see that this is recurring and deep-rooted problem. Having an accent can be a significant hindrance to establishing a career.