Yes. Using the same restriction enzyme will produce the same sticky ends, so when you mix the plasmid (vector) with the DNA fragment of interest, some of that combo will ligate together. However, other plasmids will just ligate upon itself too and so, to ensure the fragment actually was inserted, we use test markers to select for certain bacterial colonies. A good example is a plasmid with the X-gal gene. X-gal codes for an enzyme that digests an analog of lactose. A few restriction enzymes have a restriction site within this gene, so if the fragment is inserted properly, the enzyme will be defective and the colonies will appear white (indicating the fragment was transferred succesfully); if colonies appeared blue, this indicates the enzyme was functional and the fragment was not inserted. You also have to be careful because some bacteria won't take up plasmids at all (only a small percentage actually do), so another test marker typically used is some type of antibiotic resistance gene. Plasmids transfered succesfully can grow on the medium containing the antibiotic, while those missing the plasmid will be selected against.
