In your subs with an error handler, you need the red exit sub before the error handler.
Without the exit sub, the code will ALWAYS fall thru to the error handler. That's why you were getting those pesky error messageboxes when there was no error.
Code:
Public Sub cmdOpenTable_Click()
    On Error GoTo problem_Error


    DoCmd.OpenTable "tblBIns"
    DoCmd.SelectObject acTable, "tblBins"
    DoCmd.Requery
    Exit Sub
problem_Error:
    Select Case Err
        Case 7874
        MsgBox " Create New Table! "
    Case Else
        MsgBox Err.Number & ":" & Err.Description
    End Select
End Sub