Jinglebellfrocks please read 
Reality check: can owning a cat be grounds for appeal against deportation?
Is it true?
No. The original reports of the court case make clear that the appeal was actually a dispute about a rule which dictated that if someone was settled in the UK in a relationships for more than two years without enforcement action, then they automatically have a right to remain.
The cat was mentioned in the appeal and judge's decision, as evidence of the man's relationship - not as a reason in itself.
Original reports from the 2009 case also make clear that the cat was only mentioned in passing and wasn't the reason the man was ultimately allowed to stay.
The Telegraph story reports the solicitor in the case Barry O'Leary, saying the cat was only mentioned to demonstrate the couples long-term commitment to one another. The appeal was won not on the basis that the man had a cat here, but that it was one element that represented the commitment of the relationship he was in. The solicitor said:
As part of the application and as part of the appeal, the couple gave detailed statements of the life they had built together in the UK to show the genuine nature and duration of their relationship. One detail provided, among many, was that they had owned a cat together for some time.
The appeal was successful and when giving the reasons for the success the judge did comment on the couple's cat. It was taken into account as part of the couple's life together. The Home Office asked for the decision to be reconsidered. They argued it should be reconsidered because the decision was wrong in law, and one error they cited was that too much consideration was given to the couple's cat.
The home secretary mentioned the case in the context of her plans to change the rules to stop convicted criminals resisting deportations on the basis of article 8 – family ties. But the Bolivian man – who has never been named – was not even a convicted criminal.
A paper (pdf) by The Human Rights Futures Project at LSE debunked this myth in July this year. I've just spoken with one of the researchers on the paper who says their source for this information was the solicitor and barrister in the case, who approved their interpretation of events. The paper says:
This case is often listed, misleadingly, alongside cases... of convicted criminals who challenge their deportation on Art 8 grounds. In fact, the case concerned a man who came to the UK as a student and was refused leave to remain and did not concern deportation on grounds of criminal conviction. The immigration judge had allowed his appeal on the basis of a former Home Office policy (DP3/96) which said that if an individual lived in the UK with a settled spouse for two years or more without enforcement action being taken against them, they were entitled to leave to remain. The appeal was also allowed on Art 8 grounds – he had a long-term relationship with a British citizen and they had lived together for four years. The reference to the cat was one detail provided by the couple as evidence of their long-term relationship but did not form any part of the tribunal's reasons for deciding that he should be allowed to stay in the UK.47 The Home Office appealed but the senior immigration judge upheld the decision on the basis that the former Home Office policy (DP3/96), although it had since been withdrawn, still applied in this case (due to the date of the initial decision)
www.theguardian.com/politics/reality-check-with-polly-curtis/2011/oct/04/reality-check-cat-theresa-may