Vodafone will not offer data-free access to BBC Bitesize during lockdowns in the UK, saying it is not technically possible to ring-fence it from other BBC content.
Mobile companies have been asked to find ways to make home-schooling easier for families without fixed broadband.
Most, including Vodafone, have announced education resource Oak National Academy will be free.
But only BT, EE and Plusnet have said the same for BBC Bitesize.
O2 told BBC News that it was in discussions with the BBC about zero-rating Bitesize and was assessing the "technical feasibility". Three said it would not be able to offer access to the site data free, because it would require also offering access to the entirety of the BBC's website, including BBC iPlayer.
Vodafone told BBC News its technical team had concluded it was not possible to offer free access to BBC Bitesize without doing so for the whole of the BBC's UK website.
The company's general counsel and external affairs director Helen Lamprell said zero-rating educational sites "sounds simple enough to do in theory but there are a number of technical and legal challenges".
"For example, if an education-resource site hosts some of its content on another platform, such as YouTube, as is often the case, we would have to zero-rate the whole of YouTube," she said.
That is "not fair to other content providers" and would be difficult to square with "legal obligations around net neutrality".
Net neutrality, the principle all traffic should be treated equally, has been enshrined in EU law and has also been adopted by the UK, post-Brexit.
On its website, the regulator Ofcom says broadband providers "must treat all internet traffic on their networks equally, and not favour certain websites or services".
EE, BT Mobile and Plusnet Mobile - all owned by BT - have promised BBC Bitesize content will be free for its customers from the end of January.
They have also asked the Scottish, Northern Irish and Welsh administrations to each nominate an educational resource they want zero-rated.
BT told BBC News: "We recognise that process of zero-rating a site or service, is technically difficult, and potentially costly.
"However, it's not impossible and, in certain situations like this, is something we felt was worth exploring as part of our Lockdown Learning package, if only as short term measure."
Ofcom says more than 880,000 children live in a household with a mobile internet connection only.
And during the first national lockdown, in spring, there was evidence many families were burning through their data allowances.
All broadband and mobile companies are making efforts to help families struggling with home-schooling because of connection issues.
And there have been a swathe of initiatives to ease the burden.
Vodafone, for example, is giving away 350,000 free data Sims cards to disadvantaged schoolchildren.
So far, they have been ordered by more than 9,000 schools and colleges around the UK.