My dad used to have a theory that he frequently would espouse to us as
children.  Although he is now long gone, his thinking has stuck with
me.  It was as follows:

A fully grown man can pick up a newly born baby calf no problem (for
theatrical confirmation see City Slickers).  Now, presumably that calf
would not grow so much that day that the man could not pick it up the
next day as well.  In fact, the cow should never grow so much in one
day that the man, having been able to pick it up the day before,
should not still be able to pick it up.  Thus, when the cow is fully
grown, the man would still be able to pick up the cow.

However, clearly a man cannot pick up a full grown cow.  Thoughts?

Your reasoning is OK, but it is based on faulty assumptions. You
assume that the calf will not grow so much in one day that the man
won't be able to pick it up the next day, but is this valid?  A calf
can gain 2 lbs per day early on, slowing down after the first 6 weeks
or so, but still gaining appreciably.  Early on this may not seem like
much, but one day you will get to the point that it is all the man can
do to get it off the ground.  The next day the calf will be even
bigger.  Will the man have gained enough strength by the next day to
lift the now even slightly bigger calf? If so, how many days in a row
can he increase his maximum lifting ability?
If you have any experience with weight lifting, you will see that
there will be a point at which this breaks down.  A full grown cow can
weight 1000-1500 lbs.  Olympic or World class weight lifters are not
achieving this much weight.  If it were as simple as increasing the
amount you lift by 2 lbs each day, we would see this all the time. 
Instead, the world class lifters are struggling with daily workouts to
increase their maximum by even a lb or two in time for the next
competition.

Sounds a lot like the paradox that you never get home if you
constantly divide the distance by half

Sorry to be dismissive, but this is clearly a fallacy, for the reasons
set out by ac67. But why on earth did your dad pick on a cow as his
subject; why not a person? After all, a baby gowing into child- and
adult-hood  puts on even less weight each day than a calf/cow, but
every parent (except a professional weightlifter) knows that the day
will arrive when they won't be able to lift up their child.

Simply change the type of cow you are trying to pick up.  Some of
these would be a breeze.
http://www.miniaturebull.com/

No bull.

Cheers
Digsalot - whom someone recently described as eccentric??????

The legend of a child which carries a calf each day, growing gradually
strong until at maturity is able to carry the full grown animal, is
associated with a famous ancient Greek Olympic champion wrestler:

[Milo of Croton]
http://encyclopedia.thefreedictionary.com/Milo%20of%20Croton

For more about this remarkable character:

[Milo of Kroton]
http://www.perseus.tufts.edu/Olympics/milo.html

regards, mathtalk-ga

Some say this calf->bull lifting was the birth of progressive weight training.

The paradox that corwin02 mentioned is related to Zeno's paradox, a
race between Achilles and a tortoise.  It "shows" that Achilles cannot
beat a tortoise with a head start.

Similar ideas, swapping weight for distance, but neither are provably
true, as others have already pointed out.