Describe a person who brags about difficulties
Solution 1:
An activity closely related to this is sometimes called misery poker. This refers to when two (or more) people are comparing how bad they have it, playing their miseries as you would cards in poker – as in the Four Yorkshiremen sketch that Michael quoted, though in general you don't bluff (much) in misery poker, at least not to the extent the Yorkshiremen do.
The term was in common usage at Swarthmore College when I was there (2007–11). It seems at least somewhat spread beyond that: there's a tvtropes article, and you found a WSJ article. But it's not common enough in print to be in Google ngrams, my officemate who's a Brown alum from the same time period hadn't heard of it, and looking through Google I'm seeing primarily Swarthmore-related results. It seems that non-Swatties likely won't have heard the actual term, but its meaning might be sufficiently transparent to use.
There's also a new version of it that I like a lot: "misery Pokémon." I recently saw it referred to in this recent article from one of the campus papers1, but the first instance I could find was in the 2010 orientation play2, which was probably most students' introduction to at least the name of it. (I'm fairly certain I was introduced to the concept of misery poker by the 2007 version of this play, though it didn't contain misery Pokémon.) There's a video of the play on YouTube, with the misery poker component beginning here and the Pokémon part starting at 1:12:20. They also give passing references to "Trivial Pursuit of Happiness," "Torment Yahtzee," "Bitchy Bitchy Hippos," and "Apathy to Apples."
1. Full disclosure: I was once editor-in-chief of that paper, and my cousin is currently managing editor. It's a perfectly legitimate student paper, it just feels weird not mentioning that. :p
2. I was actually in the audience at that play, but I forgot about it until finding a reference to the clip....
Solution 2:
Martyr
Usually used in the religeous sense, it can also be used in a less dramatic way
A person who displays or exaggerates their discomfort or distress in order to obtain sympathy: "she wanted to play the martyr"
http://www.oxforddictionaries.com/definition/english/martyr
Solution 3:
Not sure what the word is for it, but Monty Python's "Four Yorkshiremen" sketch absolutely nailed it!
Graham Chapman: Nothing like a good glass of Chateau de Chassilier wine, ay Gessiah?
Terry Gilliam: You're right there Obediah.
Eric Idle: Who'd a thought thirty years ago we'd all be sittin' here drinking Chateau de Chassilier wine?
MP: Aye. In them days, we'd a' been glad to have the price of a cup o' tea.
GC: A cup ' COLD tea.
EI: Without milk or sugar.
TG: OR tea!
MP: In a filthy, cracked cup.
EI: We never used to have a cup. We used to have to drink out of a rolled up newspaper.
GC: The best WE could manage was to suck on a piece of damp cloth.
TG: But you know, we were happy in those days, though we were poor.
MP: Aye. BECAUSE we were poor. My old Dad used to say to me, "Money doesn't buy you happiness."
EI: 'E was right. I was happier then and I had NOTHIN'. We used to live in this tiiiny old house, with greaaaaat big holes in the roof.
GC: House? You were lucky to have a HOUSE! We used to live in one room, all hundred and twenty-six of us, no furniture. Half the floor was missing; we were all huddled together in one corner for fear of FALLING!
TG: You were lucky to have a ROOM! We used to have to live in a corridor!
MP: Ohhhh we used to DREAM of livin' in a corridor! Woulda' been a palace to us. We used to live in an old water tank on a rubbish tip. We got woken up every morning by having a load of rotting fish dumped all over us! House!? Hmph.
EI: Well when I say "house" it was only a hole in the ground covered by a piece of tarpolin, but it was a house to US.
GC: We were evicted from our hole in the ground; we had to go and live in a lake!
TG: You were lucky to have a LAKE! There were a hundred and sixty of us living in a small shoebox in the middle of the road.
MP: Cardboard box?
TG: Aye.
MP: You were lucky. We lived for three months in a brown paper bag in a septic tank. We used to have to get up at six o'clock in the morning, clean the bag, eat a crust of stale bread, go to work down mill for fourteen hours a day week in-week out. When we got home, out Dad would thrash us to sleep with his belt!
GC: Luxury. We used to have to get out of the lake at three o'clock in the morning, clean the lake, eat a handful of hot gravel, go to work at the mill every day for tuppence a month, come home, and Dad would beat us around the head and neck with a broken bottle, if we were LUCKY!
TG: Well we had it tough. We used to have to get up out of the shoebox at twelve o'clock at night, and LICK the road clean with our tongues. We had half a handful of freezing cold gravel, worked twenty-four hours a day at the mill for fourpence every six years, and when we got home, our Dad would slice us in two with a bread knife.
EI: Right. I had to get up in the morning at ten o'clock at night, half an hour before I went to bed, (pause for laughter), eat a lump of cold poison, work twenty-nine hours a day down mill, and pay mill owner for permission to come to work, and when we got home, our Dad would kill us, and dance about on our graves singing "Hallelujah."
MP: But you try and tell the young people today that... and they won't believe ya'.
ALL: Nope, nope..
Solution 4:
A good literary exemplar might be the inept clerk Semyon Yepikhodov in Chekhov’s The Cherry Orchard. He is forever bewailing his endless and daily mishaps. Dunyasha says early in Act I,
He’s an unlucky man; every day something happens. We tease him about it. They call him “Two-and-twenty troubles.”
Such a sobriquet seems eminently recyclable.