""broken up" when a relationship stops working, but "broken down" when a car stops working?"

A machine 'breaks DOWN' when it stops working

An object that falls apart (into many pieces - see 9 below), even a relationship 'breaks UP'.

~~~~~~~~~~~~~
http://www.merriam-webster.com/dictionary/UP

Function:
    adverb
Etymology:
    partly from Middle English up upward, from Old English ūp; partly from Middle English uppe on high, from Old English; both akin to Old High German ūf up and probably to Latin sub under, Greek hypo under, hyper over — more at over

Date:
    before 12th century

1 a
(1): in or into a higher position or level ; especially : away from the center of the earth
(2): from beneath the ground or water to the surface
(3): from below the horizon
(4): upstream 1
(5): in or into an upright position [sit up] ; especially : out of bed
b: upward from the ground or surface [pull up a daisy]

c: so as to expose a particular surface

2: with greater intensity [speak up]

3 a: in or into a better or more advanced state
b: at an end [your time is up]
c: in or into a state of greater intensity or excitement
d: to or at a greater speed, rate, or amount [prices went up]
e: in a continual sequence : in continuance from a point or to a point [from third grade up] [at prices of $10 and up] [up until now]

4 a
(1): into existence, evidence, prominence, or prevalence
(2): into operation or practical form
b: into consideration or attention [bring up for discussion]

5: into possession or custody

6 a: entirely , completely [button up your coat]
b: used as an intensifier [clean up the house]

7: in or into storage : by [lay up supplies]

8 a: so as to arrive or approach
b: in a direction conventionally the opposite of down:
(1): to windward
(2): northward
(3): to or at the top
(4): to or at the rear of a theatrical stage

9: in or into parts

10: to a stop —usually used with draw, bring, fetch, or pull

11: for each side [the score is 15 up]
~~~~~~~~~~
"The more you learn, the less you know"