Short answer: as early as you can. Here is exactly why, and what it means if your pet is already grown.
The single most important rule of pet insurance is this: it never covers a condition that already exists. That one rule is what makes age the most important factor in the value you get. The younger and healthier your pet is when you enroll, the more a policy will ever cover and the less you'll pay for it.
Most insurers let you enroll from 6–8 weeks old. Enrolling now, before vaccines are even finished, means virtually everything that could go wrong later is on the table for coverage. This is the ideal window.
If your dog or cat is a few years old but healthy, insurance is still very much worth considering. Premiums are higher than for a puppy, but anything not yet diagnosed is still coverable — and most pets haven't had their expensive health event yet at age three or four.
For older pets the math gets tougher: premiums are high, many conditions may already be pre-existing, and a few insurers cap the age at which you can start a new accident-and-illness policy (often around 14). Accident-only coverage usually remains available. If your senior is healthy, get quotes — but read the exclusions carefully.
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As young as possible — most insurers allow enrollment from 6–8 weeks. Enrolling while your pet is young and healthy means lower premiums and the broadest coverage, because pre-existing conditions are never covered.
Probably not for accident coverage. Some insurers cap the age to start a new accident-and-illness policy (often around 14 years), and older pets have higher premiums and more pre-existing exclusions, but coverage is often still available. Get quotes and read the exclusions.
No. Standard pet insurance never covers pre-existing conditions. Some insurers may cover a "curable" past condition again after a symptom-free period, but anything ongoing is excluded.