Yes and no.
Yes, because the genetic material in the nucleus is constantly "expressing" some genes and turning others off, and which genes are being expressed determines what proteins and enzymes are produced, which in turn regulates cell behavior and gene expression. We cannot currently control what the nucleus is doing, so cloning from a somatic cell consistes of snatching the nucleus from one cell and injecting it into an egg cell that has already had the haploid nucleus removed, then jolting it with a miniscual electrical current to sort of jumpstart things. This is basically the equvalent of throwing your shoe at the TV to change to your favorite channel. Sometimes the nucleus is able to take a cue from the surrounding cytoplasm and reset itself to work as a zygote nucleus, but more often it doesn't.
But even with improved techniques (and there is a lot of room for improvement) the fact remains that some specialized cells don't just have different sequences expressed, they may have actually lost certain parts of the DNA and thus not have a full set (some specialized cells don't have nuclear DNA at all). These types of cells simply cannot provide a nucleus for cloning. By contrast, most types of cells have a complete set of DNA.
The barrier that is currently insuperable, though, is telomere decay and the lack of controlled means of activating the telomerase reaction. Telomeres are the terminal ends of each chromosome. Every time a cell divides and the DNA replicates, both copies end up with one less telomere. All living cells have a process that reverses this decay, but in multicelled organisms this is turned off for any cells not involved in reproduction. Because the reproductive cells are rare, hard to get at, and tend to want to divide into haploid cells, it is impractical to clone from them (it would be difficult, painful for the donor, and you would just end up with a cell that would divide into two eggs instead of developing into a blastocyst the way a zygote is supposed to). Thus, this problem is more akin to...trying to record a three hour movie on a two hour tape.
Given current methods, even a successful clone (normal genetics, normal development, normal growth) will hit a wall as the cells become too old to continue dividing (which is essential for tissue repair). For instance, Dolly (the sheep) now has symptoms of aging consistent with a sheep more than twice her actual age (arthritis and other degenerative conditions). Her cells cannot continue to divide, because they've reached the division limit already (or have begun to approach it, the division limit is not a sudden stop, as the cell approaches it it becomes progressively less capable of dividing or carrying out other activities). Soon she'll be riddled with cancers as the cells at the limit begin to mutate (when the telomeres are gone, the ends of the chomosomes unravel and the cell either dies or goes renegade, rebuilding the telomeres "illegally" and breaking any other rules that seem too restrictive--I know that's an egregious anthropomorphism, what actually happens is an example of microevolution, all the cells that don't break the rules are selected against and those that do break the rules reproduce and thus predominate, there is no element of consciousness or volition).
Cancers always begin to appear once the division limit is reached, this is pretty consistent for all mammals and probably most animals generally. This is the genetic instability which is related to telomere decay, and is not a question of what type of cell the nucleus belongs to, and is the ultimate limiting factor on the lifespan of a multicellular organism.
