When drafting in the late rounds, I need to know which 53 man roster spot they are filling.  If the player can't make the team, he is worthless to us.  Using your terminology, I'm definitely one who prefers a better bite at the apple versus two bites.  If you know that you only have 1 roster spot left for a specific position and you have two sixth rounders, I try to trade the two picks for one fifth rounder.  If not, what happens?  You cut one of the sixth rounders and you are left with a sixth round talent.  With the trade, you would have a fifth round talent on the roster.

I don't see six late round picks making our roster.  How many will have a realistic shot is hard to know prior to free agency.  If I had to guess, maybe three.  For the others, I would do one of the following (listed in order of preference):

1. Trade for a veteran.  We have plenty of salary cap room.  Some teams (such as New Orleans) do not.  If it makes sense, I would rather have a veteran who can make the team versus a rookie who cannot.
   
2. Use to trade up.  Prior to the draft, this is hard to be specific as we don't know how the draft will fall and we don't what trades teams ahead of us will be willing to take.  The bad news is that late round picks don't have much value and other teams know it.  However, they sometimes can be used to "leap frog" when a quality player at a position of need falls and we are worried a team right in front of us might take him.
   
3. Trade for future picks.  Typically you can trade a pick in the current draft for one round higher in the next year's draft.  If a seventh rounder isn't going to make this year's team anyway, why not trade it for a sixth rounder next  year when we might need it?  Currently, I believe our only extra pick in next year's draft is one fifth rounder.  We might need more depth next year.
   
4. Swing for the fences.  Sometimes a pick who has higher talent falls because they are seen as risky for some reasons (maybe a major injury, etc.)  If a safe pick won't make our team anyway, I would rather roll the dice and take a chance on someone who might.
   
5. Two bites at the apple.  I guess if all fails, you take two people at the same position and see who is better.  I don't like it, but if it's your last resort, it's your last resort.  Maybe the one who is drafted second is better than the one drafted first.  Injuries might happen to people above them on the depth chart and both make the team.  Maybe you can sneak one onto the practice squad.  However, to me, it just doesn't feel like we are playing the odds here.
   

---