The first problem that presents itself is that the ancient Hebrew concepts of father and son were not limited to the individual's male parent or offspring, but applied to any male ancestor or descendent; thus you have statements like, "Jesus Christ the son of David, the son of Abraham" (Matt. 1:1), even though there were many intervening generations between Abraham and David, and between David and Christ.
This goes far beyond the mere use of the words "father" and "son," but applies to the concept of parenthood regardless of the terminology used. Case in point, the Hebrew verb "to beget" used in the Genesis genealogies (yalad) is used in the above sense, particularly in the book of Genesis: we are told that Canaan begot whole ethnic groups (10:15-18), and that Leah, Rachel, Zilpah, and Bilhah begot their grandchildren and great-grandchildren (46:6-25). So we see that the concept of parenthood, and the terms used in the Genesis genealogies in particular, can refer to any ancestor and not merely to one's parent.
In fact, the concept of fatherhood is sometimes even extended to one's predecessor regardless of whether or not they're related. Genesis 4:20-21 describes Jabal as "the father of those who live in tents and raise livestock," and his brother Jubal as "the father of all who play the harp and flute." What's especially interesting about this passage is that the context in which it occurs is concerned with biological fatherhood. Thus, the Genesis genealogies move back and forth between different concepts of fatherhood without any textual indications that they are doing so.
This leads us to another factor: Jewish genealogies were not exhaustive, nor were they meant to be understood as such. Rather, they were selective to give the "highlights," or to emphasize a numerical structure by reducing the number of names to a multiple of seven and/or ten (this is called "telescoping"). For example, when Matthew describes the genealogy of Jesus (Matt. 1) he deliberately skips over several people who lived during the exile to Babylon. Why? Because he's trying to arrange it so that there are 14 names from every period he's describing (from Abraham to David; from David to the Babylonian exile; and from the exile to Christ). This does not constitute error, it's simply the way the ancient Jews wrote genealogies. A similar pattern is found in the genealogies in Gen. 5 and 11 where each list consists of ten names, which implies that the author was limiting who he included for some literary purpose. Other examples of this include Ruth 4:18-22 which lists 10 names; 1 Chronicles 1:5-23 (the list of the nations) which lists 70 names, and Luke 3 which lists 77 names (21 + 21 + 14 + 21).
In fact, the Bible specifically tells us that there are gaps in the Genesis genealogies: in Luke 3:35-36 we are told that Noah was the father of Shem, who was the father of Arphaxad, who was the father of Cainan, who was the father of Shelah. But this genealogy is mentioned twice in the early chapters of Genesis, and in both cases Cainan is not mentioned. Instead we are told that Arphaxad begot Shelah (Gen. 10:24; 11:12):
Luke: Noah --> Shem --> Arphaxad --> Cainan --> Shelah
Genesis: Noah --> Shem --> Arphaxad ---------------> Shelah
Again, I want to reiterate that this is not a contradiction, it's just how Hebrew genealogies were written.
If we delve further, we find that there is a very close parallel to the Genesis genealogies elsewhere in the Bible: the genealogy from Jacob to Moses is described in four passages, two of which are traditionally ascribed to Moses himself (Exod. 6:16-20; Num. 26:57-59; 1 Chr. 6:1-3; 23:6-13). All of these passages list five generations: Jacob --> Levi --> Kohath --> Amram --> Moses. Part of the problem comes in when we recognize that Gen. 46:11 states that Kohath was born before the descent into Egypt, and one of the genealogies (Exod. 6:16-20) states how long the individuals lived. If we ignore the facts that Kohath was probably already a grown man when he went to Egypt, and that he and Amram probably didn't father their children on their deathbeds, the maximum amount of time the Bible gives us between Kohath's birth, before the descent into Egypt and the time when Moses led the people out of Egypt is Kohath's lifetime (133 years) plus Amram's lifetime (137 years) plus the age of Moses at the time of the Exodus (80 years -- Exod. 7:7). But this only adds up to 350 years, and the Bible states explicitly that the Hebrews were in Egypt for 430 years (Exod. 12:40-41).
This problem is further exacerbated when we note that 1 Chronicles 7:20-27 covers the same period, giving the genealogy from Jacob to Joshua, a younger contemporary of Moses. Rather than listing only five generations, this passage lists twelve. Another problem is that Numbers 3:27-28 states that at the time Moses led the Hebrews out of Egypt, Kohath had 8,600 male descendants. If Kohath was Moses' grandfather, this number cannot be correct: if Kohath had 20 sons, and each one of them also had 20 sons, and each one of them also had 20 sons, you'd still come up short -- not to mention the fact that the Bible never mentions such an extraordinary scenario. The only resolution to these problems is that there are gaps in Moses' genealogy, and we cannot create a timetable based on the ages it gives.
When the Bible intends that a series of numbers be added together to produce a total, it gives this total itself (Num. 1:46; 2:32; Matt. 1). Since the genealogies in Exod. 6:16-20 and Gen. 5 and 11 do not give any such summation, or are ever used to provide such a summation anywhere else in the Bible, they were not intended to specify the amount of time that passed during these genealogies. The ages are given rather to demonstrate the age at which someone became a father, and how long he lived, because fatherhood and old age are considered blessings from God.
So how should we understand the genealogies in Gen. 5 and 11? Given the fluidity of the Bible's concept of fatherhood and the nature of biblical genealogies, when the text says "A lived X years and begot B," it could simply mean that A was X years old when he became the ancestor of B; that is, when he fathered the genetic line that would eventually culminate in the individual B and the B tribe. To apply it to the case where we know of a gap, when the text says, "When Arphaxad had lived 35 years, he became the father of Shelah" (Gen. 11:12), it means Arphaxad was 35 when he fathered Cainan, and Cainan eventually fathered Shelah. Of course, this is ignoring the fact that there are probably more gaps between Arphaxad and Cainan and between Cainan and Shelah. Moreover, when we note that the Bible often refers to representatives of nations or tribes (like Canaan, Israel, or Judah) as if they are the individuals they are descended from and named after (Jdg. 1:3; Ps. 80:1), "A" and "B" could merely refer to representatives of the A and B clans, either ancestors or descendants of A and B.
Since biblical genealogies vary greatly in how complete they are, we simply cannot ascribe a figure to how long of a period the Genesis genealogies are meant to encompass. And since the only way to determine the age of the universe from the Bible involves calculating this figure, we cannot determine how old the universe is from the Bible.
Update (27 July): See also part 1 and part 3