One of the bigger providers of Covid PCR test kits has responded after images of overflowing drop boxes were shared online.
The firm, Randox, said it was increasing the number of drop boxes and the frequency of box collections.
The move comes after people posted pictures of returned test kits piled high on top of drop boxes in Sutton and Hampstead over the weekend.
One MP said the "pointless" tests were "becoming a bad joke".
At least one PCR test is required for all arrivals from international destinations within two days of landing in the UK.
Those returning from green listed countries, or who are fully vaccinated and returning from amber list countries, are required to take only one.
However, those not fully vaccinated from amber list destinations must take at least two tests, one on day two and one on day eight.
One of those affected, Gavin Marshall, said he had to drop off a test kit on behalf of his daughter, who had completed a PCR test two days after returning from a holiday in Portugal.
Despite the fact that the box was full, he said: "I had no choice but to leave the sample at that site, because there wasn't enough time to get to another drop box before collection deadline."
Mr Marshall called it a "Covid travel rip-off".
Rory Boland, Which? Travel Editor, said he had seen problems with several testing firms and an increase in every type of complaint. Complaints included tests not arriving on time, people being unable to book tests and misleading prices.
"It's particularly frustrating as Which? raised this with government a few months ago," he added.
"The system isn't set up for large numbers of people travelling and now many people are travelling the system is not working properly. The government might now say it is looking into the prices of tests, but it's very late in the day and these problems were foreseeable."
Labour MP Ben Bradshaw tweeted in reaction to the photos: "These pointless rip-off day two PCR tests for people returning from amber and green list countries, most of which have a fraction of our Covid-19 rates, are becoming a bad joke."
A government spokesperson said health secretary Sajid Javid had asked the competition watchdog to investigate excessive pricing and said it was working with testing providers to reduce costs.
'I was given no refund'
Nancy Diaz was due to fly to Barcelona on 21 July. When her test arrived, she registered it and delivered it to a Randox Health drop box in Edinburgh, two days before her flight.
However, Ms Diaz said she only received confirmation that her test had arrived at the company's laboratory the following evening, meaning she knew her results were not going to come through in time for her flight.
Due to the delay, Ms Diaz rebooked her flight for 24 hours later, however, when she received her negative test result at 13:26 BST on 21 July, the day she had originally planned to fly on, she was required to do another test at Edinburgh Airport as her original test results were now outside of the 72-hour window.
"I was given no refund, as they (Randox) claim the 24 hours promise is from when the test arrives at their lab, which can be whenever, apparently," said Ms Diaz.
"The fact that the test can take more than 24 hours to arrive to their lab was not mentioned."
Lizzy Merrall ordered tests for her parents and for herself on 19 July, ahead of a family holiday to Spain on 27 July.
Her parents' tests arrived, but her pre-departure test she needed to take with her on holiday to do before flying back to the UK did not.
"I contacted Randox on the 26 July asking where my tests were," she told the BBC.
"I asked them if they could at least offer me a refund for one of the tests but they never replied to me and I had to pay £80 for a last minute test."
Ms Merrrall said her test pack only arrived once she was already on holiday in Spain.
Randox has not responded to requests for comment in relation to Ms Diaz's and Ms Merrall's experiences.
A spokesperson for the Northern Ireland-based company said it was "constantly expanding and improving its Covid-19 testing capacity and associated logistics network, which is already the largest in the United Kingdom, to meet the rapidly growing demand resulting from the loosening of travel restrictions".
"Randox continues to increase the number of drop boxes across the United Kingdom, which already totals over 200, and is increasing the frequency of box collections which are already occurring multiple times per day," they added.
"Randox is providing premium testing services in dynamic and rapidly changing circumstances and is committed to continuously improving its logistics network, to ensure that international travellers receive their results in time. All of the sample kits pictured will be processed."
PCR tests are more expensive than lateral flow tests and have been criticised by some in the travel industry for pricing many potential travellers out.
The government has previously defended their use, saying they are useful as they can be sequenced to check for variants of concern, unlike lateral flow tests.
But some have criticised the government, saying they are not sequencing enough tests.
Last week, Conservative MP Huw Merriman, chair of the Transport select committee, said they should be dropped.
"PCR tests can be two to three times the cost of lateral flow testing. It's pricing out holidays for those that don't have the means to pay for those tests," he said.
"I wouldn't mind if the tests were going to be used for sequencing, so we can find all variants of concern, but if only 5% of PCR tests are being sequenced, then surely it's a complete waste of money."
The Department for Health and Social Care (DHSC) has highlighted that as the vast majority of travellers into the UK do not test positive, their tests are not sent for sequencing.
Of those sent for sequencing, a minority contain insufficient virus, but more than 95% of viable green and amber tests are sequenced, it said.
A DHSC spokesperson said: "All positive tests for arrivals from red list countries are sequenced and there is a legal requirement for private testing providers to sequence positive PCR tests for amber or green country travellers.
"We are already a world leader in genomics, with over 600,000 positive Covid-19 samples having been sequenced during this pandemic."