And another thing: if you get less colonies on Amp/Kan compared to only Amp or only Kan, then that's another indication that you have 2 plasmids. If both markers were on the same plasmid, then you'd expect the same number of colonies, but if they are on different plasmids the likelihood that an E.coli cell gets both vectors is lower than getting only one of them....
Sounds like you have a plasmid with two antibiotic resistance markers, one for kan and one for amp, your plasmid sequence and map should confirm these resistance phenotypes.
Is it possible that the competent cells already have the ampicillin resistance or the plasmid extraction is contaminated. Maybe the miniprep kit? I have never seen a kan resistance giving resistance for amp, and I have tested enough. A lot of times we get new plasmids without plasmid map and I have to test several antibiotic plates to figure out the right one. Maybe the plates with amp are wrong? Sometimes when they have been under light or the amp wtock is old, it does not work as it is supposed to.
There might be a contamination with another plasmid of other incompatibility group that posses AmpR gene- you may try to do various restriction enzyme digestions and gel electrophoresis to figure out this possibility. If restriction enzyme cleavage pattern is ok probably something different was wrong - maybe simply ampicilin (it is heat sensitive) - try freshly dissolved stock.
Have you tried making a fresh mini-prep of your plasmid after plating on Kan (without Amp?). I think it's worth doing that. If you have another plasmid with Amp-resistance as a contaminant in your plasmid stock, then selecting only for Kan and making a new mini-prep should solve the problem. The AmpR plasmid will get lost because it's not being selected for, and you should hopefully end up with a clean KanR plasmid. Transform it and see if you still get colonies on Amp plates. If you still do (hopefully less than before), just do the same thing again until the Amp plate is clean.
One more note regarding Tomasz's comment about incompatibility groups: Do your bacteria grow on plates containing BOTH Amp and Kan? If they only grow on single antibiotic plates, but not on the double plate, then it's a clear indication that you have a second plasmid in your prep and that the two plasmids are not compatible with each other.
And another thing: if you get less colonies on Amp/Kan compared to only Amp or only Kan, then that's another indication that you have 2 plasmids. If both markers were on the same plasmid, then you'd expect the same number of colonies, but if they are on different plasmids the likelihood that an E.coli cell gets both vectors is lower than getting only one of them....
a) You may have two plasmids (Digest miniprep's made from each condition (Kan+ and Amp+) and compare to make sure the insert release and the vector is of the expected size)
b) Is the competent E coli cells that you are using already resistant to either of the antibiotics (Verify by plating them on Kan+/ Amp+ plates without any plasmid transformation)
c) Another option is to figure out the source of the plasmid (commercial or not), use sequencing or digest to verify if both Kan+/Amp+ is present
Is it a shuttle vector? Some of the shuttle vectors have two resistance markers. Also, some competent cells already carry smaller vectors that are used to mantain/suppress some pathways. Is your Amp still active? We don't use Amp plates stored at 4ºC that are older than 2 weeks. Amp degrades easily and the amount left on the plate may not be enough to suppress the bacterial growth.
I have come across at least one vector that contain both ampicillin and kanamycin resistance - the pCR cloning vectors from Invitrogen (pCR 2.1, pCR II series). If your base vector is one of these or a similar vector with both antibiotics resistance, then your results are not surprising. I suggest that you look at the original vector map or sequence to confirm this.
If none of the suggestions is worked, even they are all very good, try using Carbenicillin rather than Amp. E coli may produce Beta-lactamases to break down Amp itself.
The Questions is:Is your E. coli (without the Kanamycin Plasmid) resistant to ampicillin?. If the answer is YES. You shuold use another E. coli sensible to Ampicillin to clone the Kanamycin plasmid.