Well to be fair if she jumped over the wall the garden was not fully secure.
Also it's stated quite clearly in their exclusions

As per the Section 27 of the Road Traffic Act 1998: "A person who causes or permits a dog to be on a designated road without the dog being held on a lead is guilty of an offence. It also states, in this section, "designated road" means a length of road specified by an order in that behalf of the local authority in whose area the length of road is situated - For further details see The Road Traffic Act 1988. With this in mind, a dog on a designated road must be on a collar and lead under control.

 When walking Your Pet in an area other than a designated road, You must ensure that Your Pet remains under Your control and reasonable steps must be taken to prevent Your Pet escaping onto a designated road. By way of example only, when in a field and nearing the exit/ entrance, You must ensure Your Pet is on a lead.

 You must ensure that any dog lead, collar and/ or harness is in good condition and fits Your Pet to prevent escape. You must also ensure that any lead is used in such a way as to prevent the same slipping out of your grasp should Your Pet suddenly pull away from you.

 You must ensure Your Pet cannot escape or stray from Your property and any area in which a dog is kept must be secure and appropriately fenced or otherwise secured and all reasonable steps must be taken to prevent escape. When loading Your Pet into or out of Your vehicle, You must ensure that the area is either secure or Your Pet is on a lead.

That is verys very bad luck

The insurance companies don't need much of an excuse to avoid payouts but from the information provided by "trouble" it would seem they have all they need to invoke an exclusion. Hope your dog recovers well.

But it say 'Reasonable steps' to make sure the dog is secure, and you were reasonably sure she was secure, being in a walled area. What happened was a freak accident you had no reason to know it would happen, if I were you I would appeal x

"a secure, fully supervised garden" i think the insurance company would argue that if it was secure she wouldn't have been able to jump over the wall. But it's worth a try.

Sorry to be the bearer of bad news but I do think they should have given you an explanation in the letter declining your claim. I hope she continues to recover well and you can reach an agreement with the vet regarding payment.

I agree, appeal would at least be worth a go, talking about a lot of money here. I take it no one from the company came to check your garden before you were declined?

Afraid I agree, the fact she managed to jump over the wall , is reason for them to say " not secure " .

This may sound insensative, but was the car damaged in the accident , because you could also find yourself liable for the damage to the car as someone I know did , the drive of the car claimed against the dog owner for the dent in the car.

Insurance companies get away with blue murder Should be made to give measurements, how high does the wall or fence need to be to be classed as " secure"!! OP situation is an absolute disgrace and when it comes to a huge vet bill most companies will worm their way out of it!!!! Makes you wonder why bother, sometimes

Maybe they just think some things in life are avoidable, how can they stipulate the required height for a wall or fence. My Pugs couldn't clear a 3foot wall but my Dobermanns could clear a 5foot wall with relative ease. I'm not knocking the OP I'm guilty of one of the other many exclusions, my dogs exit and enter my vehicle without being on lead or the drive being secure. I would say they are under control, totally focused on getting in the car, but if the worst should happen, like it or not it would be totally my fault.