Councillors feel residents are missing out on many Alcester treasures and triumphs because there is nowhere to house certain significant items.
At a town council meeting, Alcester Heritage Project chairman David Moulson was asked if they could use the new museum to house awards, trophies and historic artefacts in their possession.
But Mr Moulson said the money for the museum from the Heritage Lottery Fund was secured on the strict understanding that it would purely be for Roman history.
Councillors say they have many items stored away out of sight, which actually belong to the town as a whole.
Mayor Wendy Weatherhead and Councillor John Bunting both said they had many items of fascinating historical value which they could not do anything with.
District councillor Susan Juned suggested they approach Stratford Council to ask if it would be possible to use the foyer of Globe House so people could view the items.
And Mr Moulson said a purpose-built building might be the solution for showing different aspects of Alcester life and history.
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