The research was commissioned by the Parliamentary and Health Service Ombudsman, which deals with complaints about bodies including NHS England, who says the process should be simpler.
The most common reason not to complain was feeling it would be "pointless".
The Department of Health said it had improved complaints procedures.
The research found:
Parliamentary and Health Service Ombudsman Julie Mellor said the findings showed concerns were going "unheard or unaddressed".
She said complaining must be made "simple and accessible" or public services would "lose opportunities to learn from mistakes".
Complaints must go to the organisation being complained about before the ombudsman service - which looked into almost 8,000 complaints last year - can get involved.
Its role is to make the final decision on complaints about UK government departments and other public organisations, and the NHS in England.
Healthwatch, which champions consumer rights in health and social care, said its own research suggested just one in five people who had a poor experience of the NHS wrote a letter of complaint.
Its chairwoman, Anna Bradley, said: "At the moment we estimate that more than 2,000 incidents of poor care a day occur across the country's health and social care services, yet the system only hears about a fraction of them because of the sheer effort it takes for patients to make themselves heard."
A Department of Health spokeswoman said: "Listening to patients and staff is vital to improving care.
"That's why we have made it easier for people to know how to complain through clearer information, we've made hospitals legally obliged to apologise to patients when mistakes happen and introduced complaints handling as a crucial element of tougher hospital inspections."
Only a third of people who experience poor service from public bodies such as the NHS make a complaint, a survey of more than 4,200 people has found.