Is the insert isolated from the gel after the digestion of another vector, or is it raised by PCR, then digested? How long are the overhangs - to get a good digestion after PCR they should be a few bases longer than the bit recognized by the enzyme. If PCR, is the reaction cleaned up with an enzymatic clean-up kit?

If you cut your insert out from the gel, it is imperative to limit UV exposition, because if it's too long, the DNA crosslinks and is impossible to replicate in bacteria. Just quickly mark the gel with the knife, turn UV off and then cut out the marked band. 

With tricky clonings, you can also dephosphorylate your vector after the digestion. 

The insert is isolated by digestion of another vector.

If your are getting colonies without insert, some of your vector must be cutting with only one of the two enzymes.  Since you are able to cut your insert out of a plasmid, both enzymes must be good.  However, if the two sites are very close together in the vector, the second cut may not happen efficiently and you won't realize this from your gel.  I would definitely dephosphorylate the vector and do a control ligation with only the vector fragment.  You should get few or no colonies from this.  If you eliminate the problem with reclosed vector, then you can consider other issues.  Damaged ends of DNA as mentioned above, poor recovery of DNA from the gel, and low efficiency of transformation are some of the most common problems.

Yes, as mentioned above. I'd suggest: 1) keep UV exposure to minimum when cutting from the gel; 2) dephosphorylate vector (and clean it up after the reaction - you will probably need to use more vector) and follow instructions above. 3) If the digestion is not efficient, do it overnight, to ensure that it happened.