A gaming system and method which provides and tracks bonus sequence eligibility points accumulated by players playing the gaming devices in the gaming system. The gaming system maintains, for one or more players, a plurality of progressive awards, wherein for a specific player, one or more of the progressive awards is based, at least in part, on the quantity of accumulated bonus sequence eligibility points associated with or accumulated by that player. If a player forfeits a bonus sequence eligibility point (due to player inactivity) or otherwise loses a bonus sequence eligibility point (due to not winning a provided bonus sequence), the gaming system contributes at least part of a value of such forfeited or lost bonus sequence eligibility points to increase a bonus sequence award maintained by the gaming system.