Is there a case where the consumer surplus being negative does not mean that the producer surplus increases? Is it true that as consumer surplus increases, producer surplus decreases and vice versa?
Consider two situations:
(i) a price so high that no consumer buys a good,
(ii) the market price at which supply and demand are equal.
CS=PS=0 in situation (i) and CS>0 and PS>0 in situation (ii).
If you move from (i) to (ii) CS increases and PS as well.
If you move from (ii) to (i) PS decreases and so does CS.
If consumers cannot be forced to buy, the CS is non-negative.