I think the formal answer is no. It is not logical. Any two species always have a common ancestor but the common ancestor almost by definition cannot be alive today. Having said that, it is possible a modern form (species) may look a great deal like the common ancestor. But both it (the one that looks like the common ancestor) and the species in question have both undergone change since the time of divergence. The rate of change is about 1% of nucleotide sequence per million years if the region of DNA under consideration is not encoding something that is under active natural selection.

So, consider two modern species: A, the one you are interested in, and B, the one that looks a great deal like the common ancestor and from the fossil record we know existed 10 million years ago. The two species will be about 20% nucleotide different in the unselected parts of the genome, such as in big introns or in large intergenic regions. Each will have changed by about 10% in base sequence since the common ancestor. The one that looks most like the common ancestor has still changed at the DNA level, you just may not be able to see it on the surface. So with starfish for example, some look pretty much like those in the fossil record from hundreds of millions of years ago, but if we could get the DNA from the ancestor, it would be quite different from the DNA in modern starfish.

For a thorough discussion I recommend the book: The Greatest Show on Earth By Richard Dawkins (Simon and Schuster, 2008). It's a very good read.

Well, the answer is more like yes but no. Yes, you can have 'living fossils' that haven't changed in appearance for millions of years while other species have branched off in new directions, but of course evolution is always happening, so that 'primitive' lineage is also changing at the DNA level even if it retains ancestral features that make it look 'less' evolved. Lampreys are a good example. These primitive, eel-like fish are one of the only jawless vertebrates alive, so all fish, not to mention all the other vertebrates (including us) evolved from a lamprey-like ancestor hundreds of millions of years ago, and yet these creatures look as if they haven't changed morphologically for at least 360 million years. For reference, check out this article in the journal Nature.