Are there ion combinations which can't be separated using the ion exchange resin method?
Ion Exchange resins are not selective.  You are essentially just statistically overpowering the situation by providing many more times the concentration of the ion you want compared to the ion you are 'extracting'.

There are some situations where chelation is so strong you can't exchange because the other ion is less stable in said position, say for example in crown ethers.  

So the answer to your question, is no, there are not specific ions that can't be exchanged as a general rule, but there are ion pair combinations which won't exchange.