Why doesn't Sanger (fluorescent) DNA sequencing double count nucleotides?
In chain-termination sequencing, a population of molecules is detected as opposed to a single one. The readout looks like this:


[ http://seqcore.brcf.med.umich.edu/doc/dnaseq/trouble/badseq.html ]

The peaks represent intensity of the four different fluorophores at the detector. Every fragment that is terminated at the same spot will have the same fluorophore and the same length. Because capillary gel electrophoresis separates molecules by size, these fragments will pass through at approximately the same time. "Approximately the same time" is why you see broad-ish and overlapping peaks on the readout as opposed to sharp and distinct lines. 

Ideally, the readout will appear like the left half of the image, where there's minimal overlap between peaks. However, often you will see overlapping peaks, as in the right half of the image. The link above explains many reasons for why this occurs. Whether or not a peak is correctly assigned is determined by Phred scores, which is a statistical analysis of the shape and resolution of each peak.