Dear Dave,
Sorry for the delay.

To clarify what is in the 'HAT' column,
run the simple job

 Test hat value
 TRT 2 Y
hat.asd
Y ~ mu


with data

1 12
1 13
1 32
1 42
1 38
1 35
2 29
2 25
2 19
2 15

The residual variance is 118

The .yht file contains

  Record        Yhat    Residual         Hat
       1      26.000      -14.00       11.80    
       2      26.000      -13.00       11.80    
       3      26.000       6.000       11.80    
       4      26.000       16.00       11.80    
       5      26.000       12.00       11.80    
       6      26.000       9.000       11.80    
       7      26.000       3.000       11.80    
       8      26.000      -1.000       11.80    
       9      26.000      -7.000       11.80    
      10      26.000      -11.00       11.80    


For this simple design,
  X  is a vector of 10 ones
  R  is  118 I

Calculating W Ci W'  (page 40 of the release 2 User Guide)
   is a matrix with every value  118/10  which is 
what is reported in the Hat column.

In the simple case where R  is scaled identity,
the variance can be factored out.  On the basis that
W(W'W)^{-1}W'  is idempotent, its diagonal elemts
should always be LE 1  which is what you expect.

In the more general case I believe that also applies.

Hence, if you have hat values greater than the residual
variance, that suggests a problem.  If you can send me an
example, I can investigate it.