An “inconsistency” in the Bible is a statement that we know does not accurately reflect things that we see outside of the Bible. But it can’t really be called a Biblical “contradiction” unless the Bible itself makes a contradictory claim somewhere else.
For example, here is Leviticus 11:6 (from the portion of the Bible describing dietary requirements which are honored primarily only by Jewish people in the 21st century)-
The rabbit, though it chews the cud, does not have a split hoof; it is unclean for you.
The problem is that rabbits don’t chew a cud. They chew their dung because their digestive system does not get all of the nutrients out of the food that they eat during the first pass. The scientific term for this is “refection”. (This behavior is an example of the inefficiencies of evolution. But it’s surely something that an Intelligent Designer would not have allowed.) I classify this is an “inconsistency” rather than a “contradiction” because the Bible doesn’t have any passages that say that rabbits don’t chew a cud.
To be fair, Christian apologists have addressed this passage. Here is a sample explanation:
"In modern English, animals that ‘chew the cud’ are called ruminants. They hardly chew their food when first eaten, but swallow it into a special stomach where the food is partially digested. Then it is regurgitated, chewed again, and swallowed into a different stomach. Animals which do this include cows, sheep and goats, and they all have four stomachs. Coneys and rabbits are not ruminants in this modern sense."
However, the Hebrew phrase for ‘chew the cud’ simply means ‘raising up what has been swallowed’. Coneys and rabbits go through such similar motions to ruminants that Linnaeus, the father of modern classification (and a creationist), at first classified them as ruminants. Also, rabbits and hares practice refection, which is essentially the same principle as rumination, and does indeed ‘raise up what has been swallowed’. The food goes right through the rabbit and is passed out as a special type of dropping. These are re-eaten, and can now nourish the rabbit as they have already been partly digested.
That explanation does not stand up to scrutiny as the comments below regarding that passage show:
"Gerah," the term which appears in the MT means (chewed) cud, and also perhaps grain, or berry (also a 20th of a sheckel, but I think that we can agree that that is irrelevant here). It does *not* mean dung, and there is a perfectly adequate Hebrew word for that, which could have been used. Furthermore, the phrase translated "chew the cud" in the KJV is more exactly "bring up the cud." Rabbits do not bring up anything; they let it go all the way through, and then eat it again. The description given in Leviticus is inaccurate, and that's that. Rabbits do eat their own dung; they do not bring anything up and chew on it.
When trying to understand a passage, one of the considerations to use when deciding between multiple possible interpretations is whether the passage could have been written more clearly if one interpretation rather than the other was the original intent of the author.
Here is that passage in context (all passages are from Leviticus 11):
 You may eat any animal that has a split hoof completely divided and that chews the cud.
 There are some that only chew the cud or only have a split hoof, but you must not eat them. The camel, though it chews the cud, does not have a split hoof; it is ceremonially unclean for you.  The coney, though it chews the cud, does not have a split hoof; it is unclean for you.  The rabbit, though it chews the cud, does not have a split hoof; it is unclean for you.  And the pig, though it has a split hoof completely divided, does not chew the cud; it is unclean for you.
Simply put, if the author of this passage really did actually understand that what rabbits do is different from what cows, sheep and goat do, why even mention rabbits immediately after those other animals? Camels, for example, really do chew a cud just as cows and goats. If any sort of distinction was really needed for rabbits, since Hebrew has a perfectly good word for “dung”, then why not say something like:
“Rabbits, because they eat their own dung, are unclean and should not be eaten”?
Doesn’t that make much more sense? Surely finding an animal that eats its own dung as “unclean” for humans to eat requires no additional explanation. The apologist argument, at best, diminishes the Bible by claiming that the book isn’t clearly written.
(Coneys are animals that don’t really chew a cud as cows do, though they don’t eat their own dung either as do rabbits. Instead coneys have multi-chambered stomachs and a complex system of digestion, elimination, and re-ingesting that enables them to extract nutrients from their diet of coarse leaves and grass. This “recycling” of fermented waste, though slightly different from that of cows, is similar enough that it is easy to justify the biblical allusion to “cud-chewing.”).
More importantly, even if the “Answers in Genesis” explanation is to be accepted, this passage highlights the difficulties of reading any part of the Bible in order to properly understand it. According to the apologists, you can’t begin to get the proper meaning from this passage unless you understand the obscure, ancient Hebrew word “Gerah”. The passage could surely have been written more clearly. Why wasn’t it?
This is an important point. If this detailed level of analysis must be applied to passages such as this, why isn’t it necessary to apply that same level to every single passage in the Bible?
Did the authors of the Bible only use obscure Hebrew words with nebulous meanings in those passages that contradicted what we see in the natural world? Of course not! Biblical apologists would have us believe that the Bible contains a “clear message” – except in those places where it appears that there are contradictions and inconsistencies. In those passages – and no where else – an understanding of ancient Hebrew is required to properly understand each passage.
That is nonsense, of course, and demonstrates an inconsistency in the Bible all by itself.
 http://www.answersingenesis.org/creation/v20/i4/rabbits.asp, referenced on November 10, 2008
 http://www.infidels.org/library/modern/jim_meritt/bible-contradictions.html, referenced on November 10, 2008
 The “coney” is another name for the hyrax or rock badger
 http://en.wikipedia.org/wiki/Hyrax, referenced on November 10, 2008