Are there ion combinations which can't be separated using the ion exchange resin method? I am currently investigating the uses of ion exchange resins (both cation and anion resins). I have not come across any ion combinations that cannot be separated using this method, are there any exceptions or does this method work for all ion combinations? If there are ions that can not be separated using the resin, what methods would work to separate them?
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.