That behaviour makes it seem to me that both layers have the same data source (ie, the same feature class), but with different query definitions. In this case, you would be copying/pasting to the same feature class (even though a different layer), but the definition queries would make sure that the copied feature will only appear in the layer from which it was copied.
To get around this, you would need to know what the definition queries on the layers are, and you would need to edit the pasted feature to match the definition query of the destination layer. You can see the definition query in the layer's Properties window.
Eg, if the two layers had the following definition queries:
Source Layer: NOTINUSE = 'N'
Destination Layer: NOTINUSE = 'Y'
Then in the layer's attributes table you would need to change the value of NOTINUSE from N to Y for a feature to cause it to disappear from the source layer and appear in the destination layer. (Or use the Attributes pane, instead of the Attributes table.)
It's worth noting that this is a common problem. Not just with copying features, but I often have the issue when creating new features and with appending features ('Append' tool). It still catches me off-guard and scratching my head sometimes, even after experiencing the issue many times. The feature(s) are created, but then disappear from both the map and the attributes table. Or the 'Append' tool finishes running successfully, but the features never appear on the map or in the attributes table. It can be very confusing until you remember that the layer has a definition query that is excluding them.
