How has it been a net gain for you; it really hasn't. IMHO a rainy day savings should more than cover 5k (mine is personally 4 months of my income). That was your dog being ill what if god forbid you or your partner come down with cancer, and you can no longer work? this is where the little planning I mentioned comes in. While emergencies are hard to plan for, they can be pre-paired for & I would argue far more valuable than the disposable income for figures.
Sometimes bad things happen, pill up and wipe out that savings. Everyone's situation is different.