The one exclusion that's virtually universal among pet insurance companies is pre-existing conditions. This means that any problem that your pet has shown symptoms of prior to purchasing an insurance policy or that manifest symptoms during the waiting period prior to the effective date of the policy won't be covered. Waiting periods for accidents and injury usually vary from none to 5 days. Waiting periods for illnesses usually vary from 15 to 30 days.
However, there may be longer waiting periods for certain specific problems e.g. anterior cruciate rupture, intervertebral disk disease, etc. The conditions and waiting periods vary from company to company. You should read a sample policy if it's available or ask a company representative what waiting periods apply to their policy.
Insurance companies go by when you first noticed symptoms, not necessarily when it is diagnosed and treated by a veterinarian. When you apply for coverage, most companies will ask you questions about your pet's medical history. Some may require a copy of your pet's medical records and/or a physical exam by your veterinarian. Some pet owners may be tempted not to tell everything - thinking that if the insurance company doesn't know about a problem, then it will be covered. More than likely, if the insurance company doesn't ask for a copy of your medical records when you apply for coverage, they will when you submit your first claim. If the insurance company finds out you've not been truthful with them, they will cancel your coverage (it's in the fine print of almost any policy).
If you indicate a previous problem, the company may tell you that specific problem won't be covered and for how long. Some policy language states that if a previous problem hasn't manifested symptoms in 180 days, then it will be covered. However, if the problem is considered chronic, it likely won't be covered at all. In fact, some chronic problems e.g. Cushings Disease or Diabetes may disqualify your pet from any illness coverage. In this case, your pet would still be eligible for accident/injury coverage.
Therefore, the best time to get insurance for your pet is while he or she is still young before any problems appear. If you know that your pet has had previous problems, try to get the insurance company, when they issue a policy, to state in writing what problems aren't covered and for how long by asking for a "medical record review." Most companies will do this if you ask, but you'll usually have to ask. Hopefully, this will prevent surprises from denied claims down the road.