Most sequencing problems are related to dirty and/or poorly quantified template.
Use high quality DNA
Generally precipitating the DNA, washing the pellet, and resusupending the DNA in ddH2O is sufficient to remove contaminants.PCR purification kits and Exo Sap will also remove some contamination.
The ratio of OD260/OD280 should be between 1.8 to 2.0.
Acurately quantify DNA
The amount of DNA used in the sequencing reaction is crucial. Too little DNA might lead to weak sequences, but too much DNA will saturate the capillary system of the sequencer, add background to your and others samples. Your sequence might also result shorter than expected, as all fluorophores will be used to create short products.
Follow the amount according to your DNA type and length, under Protocol BigDye 3.1