OP-AMP Positive/Negative Feedback together
See my answer here for an example of why you might want to use negative and positive feedback at the same time. The 598.3K resistor in the positive feedback path maintains a constant current through the variable resistor and the negative feedback path determines the gain (output volts per ohm of resistance of the variable resistor). 



To see how this yields a constant current, consider Scott's generalized NIC answer. This is also a NIC- creating a negative resistance to cancel out the incremental effect of the 100K resistor R2, so the current remains constant. 



Looking into the non-inverting input of the op-amp (without R2 and the variable resistor connected, but with R4 in place), the resistance looks like: 



Referring to the NIC schematic, we have:-

R1 = 100K || 25.68K = 20.43K
R2 = 122.2K 
R3 = 598.3K 

\$ R_{IN} = \$ - 598.3K \$ \cdot \$ \$ 20.43K \over 122.2K\$
= -100K, which exactly cancels out the effect of the 100K resistor R2 (R2 on the original schematic).