Premium bonds are a National Savings success story with nearly £35bn invested in them but many small investors say they feel excluded.
Nine billion pounds worth of prizes have been issued since 1956
National Savings & Investments (NS&I) celebrated 50 years of premium bonds by making five people millionaires in a special draw in December.
A television advert in the run-up to the event claimed: "Opportunities like this don't come along very often and when they do you have got to grab them."
But not everyone found it easy to respond.
Theresa works for the NHS in Birmingham where she has always lived.
She told BBC Radio 4's Money Box: "Sir Alan Sugar was on the TV all the time advertising the bonds. It was a good form of saving. It was a minimum of £100 and that was all I was investing."
But she was scuppered from buying bonds because she could not supply the necessary documents to prove her identity.
NS&I requires a passport and driving licence to run identity checks, but Theresa had neither so her cheque was returned.
She was very disappointed. "It was very upsetting," she said, "I have been paying taxes for 40-odd years but as far as they were concerned I didn't exist."
When challenged on the issue, Peter Cornish of NS&I said: "Like all banks and building societies we have to check the person's identity.
"The guidance in the regulations says that normally people should provide a passport or a driving licence and so on."
But he denied this would prevent people in Theresa's position from applying.
"When we write to the customer asking for the paper documentation we say if you have any difficulty providing any of this please give us a call and we can see what we can do," he said.
"Most of the checks we can do electronically. There are a small minority of people who don't have a driving licence or travel abroad and... we will ask for different identification and then the application can proceed," he added.
But that is not what appears to have happened in Theresa's case.
Asked if NS&I would look again at the rules he said: "We look at them all the time. For example, over the summer we reduced the number of paper documents from three to two, because we realised that customers may not be able to provide all three."
He said he was sorry to hear about Theresa's experience and would investigate.
But that is not the only concern the programme has heard.
The bonds pay no interest but investors can win tax-free cash prizes of up to £1m.
A prize draw is held every month with two £1m jackpots plus over a million other tax-free prizes.
Each bond has an equal chance of winning but the more bonds people have, the greater their chances.
And as individuals can invest up to £30,000 this has led to concern the competition favours wealthier people.
Money Box listener Catherine certainly believes the competition is "a haven for very comfortably-off people".
"Your chance at £100 is extremely miniscule, your chance at a £30,000 holding is much greater," she said.
"This goes against the concept of the 50 year anniversary that everybody had a chance," she added.
HAVE YOUR SAY
If the threshold was lowered, many more people could participate
But Peter Cornish denied the bonds were just for the rich.
"One of the anniversary million pound prizes was won by somebody who had just over £1,000 invested," he said.
"Every month - and you can see this on our website - you have people from just a couple of hundred pounds winning £25,000 this month, right up to those who hold the maximum of £30,000.
"And we do see customers with relatively small amounts winning high value prizes."
However he admitted it was clear "the more bonds you have the more chances you have of winning".
He said the maximum holding is often reviewed, and "when we increased it to from £20,000 to £30,000 several years ago we did seriously consider the effect that that would have on prizes.
But he said: "We feel it is a balance. People see it as real investment and there are now about 300,000 people who hold the maximum.
"The prize fund is now 3.4%. The odds remain at 24,000 to 1 which means that someone with the maximum could receive up to 15 prizes a year."
BBC Radio 4's Money Box was broadcast on Saturday, 6 January 2007 at 1204 GMT.
The programme will be repeated on Sunday, 7 January at 1502 GMT.