Thanks to peg and mac for replying to my post.
peg
I have tried four different crystals, FOX, ECS,Citizen, and Oscilent. They all had the same problem so I don;t think it is just a bad batch of crystals.
mac
I have tried 100K for Rs and one time even went as low as 47K (just for the experiment only). The problem did not go away. The reason I chose Rs=330K is that it is the one shown in the monitoring mode configuration in the data sheet.
As far as I can tell, the problem didn't seem to root in the crystal circuit (although the effect pointed to it). Just yesterday I played around with it and discovered something very interesting.
The reason that I could not start the circuit by applying a reset signal because the RST line was already held LOW by the micro-controller. This meant that the micro-controller never got out of POR state when it had the problem. To verify this, instead of a LOW signal, I applied a 5V to the RST line. As soon as I removed the 5V, there was a transition from HI-LO on the RST line and the micro-controller started to work.
All of this tells me that for some reason, the processor entered the STOP mode before it finished the POR state. This well explained the randomness of the problem because the chance the micro-controller jumps into the STOP mode at the power up is pretty low. This also explained why a LOW signal on the RST line did not reset the micro-controller because to get out of the STOP mode, it requires a transition HI-LO on the RST line.
At this point, I suspect the power rise in my circuit may be too fast for the internal LVI to properly reset the micro-controller at the power up and allows it to enter undetermined state in which by some rare chance the micro-controller executes a STOP instruction. (I have no codes in my program that put the micro-controller in the STOP mode.)
I have a 220uF filter cap in the power circuit. When I measured the power rise of my circuit, I got 0 - 5V in 4ms. Does any one know if this is too fast for the built-in LVI?
Thanks.