It is usually offered in good faith, and it is usually far too early.
The problem is timing, not honesty. An offer made in the first days is calculated from the bills that exist in the first days. Dog bite costs have a habit of arriving later: infection that needs a second course of treatment, nerve symptoms that take weeks to declare themselves, and scar revision surgery that can be staged over years — particularly for a growing child.
Once you accept and sign, it is generally finished. If the wound then costs five times what was estimated, that is your problem.
Three things worth separating:
A neighbour handing you cash is not a settlement unless you signed something. If you signed anything, get it looked at.
A payment from their insurer’s medical payments coverage — often ,000 to ,000 — is a normal no-fault provision. Accepting it does not usually resolve the claim, but confirm that before you take it.
A release is the document that ends everything. Do not sign one before you know what your treatment actually costs.
Nothing here means you must be adversarial with a neighbour. The money almost always comes from an insurance policy, not from them personally.