The key ruling, concerning refugees' access to 'adequate' housing provision, turned on the couple's persistent attempts to secure suitable accommodation to meet their children's acute needs.
As a result they need sufficient space and specially adapted accommodation to meet those dire needs, but fell victim to a gap in the law which leaves them in 'wholly unsuitable' accommodation.
The family arrived in the UK three years ago, finding initial accommodation in Hackney, before being transferred to Walthamstow. The home secretary at one time considered 'dispersing' the family to be re-housed in the provinces, but finally decided against this in April last year.
Giving judgement today, Lord Justice Waller said there seemed 'little doubt' that the children's disabilities were 'getting worse'.
There was also evidence from health experts that their current home in Walthamstow was 'unsuitable for the two children's medium and long-term needs'
The family's counsel, Alison Foster QC, said their cramped two-bed house was 'wholly unsuitable' for children with such pressing needs.
Today's appeal stemmed from a High Court ruling in June in which Mr Justice Keith ruled that neither the Waltham Forest LBC, nor the Home Office's National Asylum Support Service, was obliged to do more than provide 'ordinary' accommodation.
That ruling left the family unable to secure 'adequate' housing, such as a bungalow complete with lift, designed for disabled living.
Lord Justice Waller said he found the case 'very troublesome' in relation to the two children, but added: ' My conclusion is that NASS are not at this stage in breach of their duty.
'Steps - indeed urgent steps - must continue to be taken by NASS to see Waltham Forest do all that is reasonable to provide this family with more suitable accommodation.
'But in my view the time has not arrived when it can be said that NASS are in breach of duty under the 1999 Immigration and Asylum Act.'
The judge in turn dismissed the family's challenge to the stance taken by Waltham Forest LBC on grounds that the family was still NASS's primary responsibility.
As a result 'no remedy' was available against Waltham Forest.
Lords Justice Brooke and Clarke agreed that the appeal should be dismissed.
STRAND NEWS SERVICE