To be 'sick of something' can only be negative. It means that you have done - or been exposed to - something so much that you can't take any more. You are now thoroughly bored with it.
Another idiom with a similar meaning is 'fed up with'. If you're sick of French or fed up with learning French, you're tired of it and don't want to do it any more.
Both phrases might suggest a feeling of nausea from overeating.