Looks like we have some insurance folks in the house. Nice.
Full coverage is a mis-nomer. There is no such thing, or it's so ambiguous to be meaningless. What is "full" to your insurance carrier may mean something completely different to another carrier. Most people consider it comp (usually a mandatory coverage) and collision (an elected coverage). However, it can mean from minimum coverage (to include collision), to maximum limits with an umbrella. It's meaningless in the insurance industry.
We as consumers are responsible for what we buy and don't buy. To think an insurance company owes you something because you didn't purchase that coverage is asinine, maybe even asiten.
PIP or Medpay is first-party coverage for injuries. Every state is different so certain coverages are mandatory and others are not, but it's our responsibility as consumers to know what we pay for, or don't.
Most people are so concerned with premiums/rates that they don't even know what they buy, as long as it's cheaper than the last place. Talk to your agent (if you have one) and read the policy (declarations page). Also know that motorcycle coverages are not necessarily the same as auto. For example in my state PIP is mandatory on auto policies, Medpay is not on motorcycle policies. So, if you have auto coverage you have PIP, because you can't have auto without it. On a motorcycle policy you can purchase Medpay, but since it's not mandatory (and raises the premium) most people don't purchase it.
When your medical insurance carrier sends a letter to your auto/moto insurance carrier (if you don't carry PIP/Medpay) it's only to get a response that shows no coverage, a denial of coverage. It's because the insurance carriers need to determine which is primary (which carrier pays first, or at all). It's the normal procedure, to determine which carrier(s) should be receiving and paying bills and then if there is another carrier that steps up when that policy reaches its max.