There are several things to know about cedar chest storage of linens. For one thing, older chests can be safer than newer ones because the aged wood secretes less oil and acid. Even so, no cedar chest is safe for direct contact with linens. At the very least, there should be a barrier of muslin between linens and the wood. This barrier is probably not enough to protect linens from damage if left stored for 20 years, but it's probably enough if the chest is opened and the muslin barrier is replaced every year or so.
It is the folds of a quilt that are most likely to be damaged if left in contact with the wood. In the "old days", quilts were routinely removed from cedar chests for use on the beds during the winter months. At least they got relief from contact with the wood for half of every year! Plus, when they were stored again chances are that the fold lines changed and different areas of the quilt were in contact with the wood. This would have minimized any damage from direct contact with the cedar.