Short answer: Run-in claims were incurred before the new policy started but paid during it; run-out claims were incurred during a policy but paid after it ends. If the old policy has no run-out and the new one has no run-in, a large claim incurred in December and paid in February can be covered by neither.
Medical claims are often paid weeks or months after the service, and a stop-loss policy covers only claims that meet both its incurred and paid windows.
When changing stop-loss carriers, line up the contracts so every claim has a home: either the expiring policy runs out long enough (for example a 12/15 or 12/18), or the new policy picks up run-in (for example a 24/12 or 15/12). Confirm in writing that the new carrier accepts the earlier-incurred claims and on what terms; overlap does not allow double recovery.
Watch large ongoing cases: a new carrier may laser known claimants or limit run-in claims for them. Get both carriers’ terms side by side before giving notice.