What results from the addition or deletion of nucleotides in any number except a multiple of 3?

Addition or deletion of nucleotides in any number besides 3 results in a "frame-shift mutation."
This is because every 3 nucleotides of DNA/mRNA exons codes for a single amino acid in the synthesis of a protein. This triplet codon theory means that if 3 nucleotides are added or deleted then an amino acid will be added or lost, but subsequent codons and amino acids will still be read correctly.
However, if any number of nucleotides other than 3 are added or removed, then the codons following the mutation will be out of "sync," in terms of the reading order.
Hence it is called a frame-shift mutation because it shift the reading frame when translating nucleic acids into proteins. Frame shifts will lead to the wrong amino acids being adding in the wrong order for the rest of the code after the mutation.