Organs from 505 registered donors could not be made available for transplant in the last five years because of objections from relatives.
BBC 5 live found that almost a third of families blocked organ donation because they felt the process took "too long".
The law states that consent lies with the deceased, but in practice, relatives' wishes are always respected.
The NHS wants to reduce the number of "overrides" by encouraging prospective donors to talk to their relatives.
In England, NHS figures showed that 457 people died last year whilst waiting for an organ transplant.
Rachel, 17, from Stoke-on-Trent, wants to be an organ donor, but is concerned that her family do not support her wishes.
She told 5 live: "I wasn't aware when I signed up that your family had to be supportive of your decision. It seems like, well, what's the point of signing up if it could be overruled anyway?
"It does worry me because, if I died now, my mum does make the main decision. I hope I can trust her to make the right one."
When somebody dies who is on the Organ Donation Register, specialist nurses from NHS Blood and Transplant work with their family.
If relatives object, nurses will encourage them to accept their loved one's decision, and make it clear that they do not have the legal right to override it.
However, in practice, if a family still refuses, the donation does not go ahead.
Ben Cole, a specialist nurse for organ donation working in the Midlands, said it was "frustrating" when families say no.
"We understand that families are approached about donation at a very difficult time, and it can come as a shock to find out their relative had made the decision to donate.
"I had one family whose son had joined the Organ Donor Register, but they found it hard to believe because he'd never spoken about it.
"Another family said their dad would have ticked any box, and so weren't convinced he'd signed up intentionally.
"The relationship we build with a family at this time is so important, particularly as they can provide vital information about their relative before donation.
"If they are strongly opposed to donation, we would not want to upset them further."
Other reasons relatives gave for refusing consent include that they thought "the patient had suffered enough", they "didn't want surgery to the body", or the family were divided over the decision.
Anthony Clarkson, assistant director of organ donation and transplantation for NHS Blood and Transplant, said: "Although the number of blocked transplants is declining, a number of families each year feel unable to support their relative's decision to be a donor.
"As a result hundreds of opportunities for potentially life saving transplants are being missed every year."
There are currently 6,406 people on the transplant waiting list across the UK.
Jess Harris, 29, from London, needs a pancreas and a kidney. She thinks it's a "crazy system" that gives families the final say.
"Why isn't it like your will? Why don't they have to honour your wishes?" she told 5 live.
"I don't know why anyone would be against donating organs - one person can save up to eight lives and you're not going to need them when you're dead."
But Dr Rebecca Brown, a research fellow in practical ethics at the University of Oxford, supports families having the final say.
She says: "There's an implication that these families are selfish or unreasonable, but I don't think that's the case.
"Losing a loved one, in sudden circumstances, is very traumatic and forcing them to go along with organ donation when it is something to which they feel strongly opposed, would be very distressing.
"This is a relatively small number of families and going against their wishes would be frankly awful for them and would create all sorts of problems."
In 2016/17 the total number of deceased donors was 1,413. In the same year, families blocked the donations of 91 people who had signed the register.
In December 2015, Wales adopted an opt-out system of organ donation, but families can still have the final say over their loved one's donation. Last year, nine people in Wales who had signed up to the organ donation register were blocked from donating their organs.
Prime Minister Theresa May has pledged to introduce presumed consent for organ donation in England and a consultation will be held before the end of the year.