Why is it that if I count years from 2011 to 2014 as intervals I get 3 years, but if I count each year separately I get 4 years? I'm not a very smart man. I'm trying to count how many years I've been working at my new job. I started in May 2011.
If I count the years separately, I get that I've worked 4 years - 2011 (year 1), 2012 (year 2), 2013 (year 3), 2014 (year 4).
But if I count the years as intervals, I get that I've worked 3 years 2011 to 2012 (year 1), 2012 to 2013 (year 2), 2013 to 2014 (year 3).
Have I worked 4 years or 3 years?
 A: For a short analogy: I was born in 1986, so I am 27 now. But I have lived in four different decades (the 80s, the 90s, the 00s and the 10s). This is not the same as forty years.
In the same way, you worked in four different years, but not for four years.
A: You're driving along a highway, starting at mile marker 11.  After a while you get to mile marker 14.  You've driven 3 miles, but you have seen 4 mile markers.  To count miles traveled, count the mile marker at the end of each completed mile.  You completed miles at markers 12, 13, and 14.  
On the same trip you noticed that telephone poles are 100 feet apart.  You start counting phone poles.  How many poles do you count from pole number 11 to pole number 14 (inclusive)?  How far did you travel?
A: When I studied Discrete Math at uni, they referred to this principle as "$1$ rope has $2$ ends".
The analogy in your example is a rope whose length is $3$ years, which has $4$ marks on it:

A: You started in May 2011.
By May 2012, you had worked 1 year.
By May 2013, you had worked 2 years.
In May 2014, you completed 3 years.
It won't be 4 years until May of 2015.
A: You've worked 3 years. The "interval approach" is the correct approach:
May 2011-May 2012. 
May 2012 - May 2013.
May 2013 - May 2014.
May 2014 -  July 27 2014.
= Three years and anywhere from two to three months, depending on your start date in May 2011.!
Unless you worked all of 2011, all of 2012, all of 2013, and all of 2014, you cannot claim 4 years. In effect, working four years would require having worked the entire year of 2011, 2012, 2013, and 2014,  meaning you would have worked: January 01, 2011 until December 31st, 2014.
In reality, though, you'd need to subtract from four years the first $4+$ months of 2011, as well as the last 5+ months of the current year.
A: You worked 3 years between May 2011 and May 2014, and 2 extra months. 
Note that in 2011 and 2014 you only work a part of the year, you should not count them as full years (as you do in the first count).
A: I see the problem of how long you have been working has been solved by others. Now let's focus on what you are asking in the title to your question.
The reason you get different numbers, is because you are counting 2 different ways. If you truly want to know how long you have worked, you have to think in terms of completed units (years, months, days, etc) to however much detail you are interested in.
Once you start thinking of completed years, 2011 is no longer contributing to the counting. You have completed 1 year in May of 2012 (As @amWhy noted), 2 years in May of 2013, etc.
When thinking in completed years, the amount of years will be
a) the difference between the end and start year (provided you have passed your hire date in the end year)
b) the result from a) minus 1 (provided you have yet to pass your hire date in the end year)
When you count by intervals, you are getting the difference between 2014-2011.
When you count the years, you are counting the numbers in a sequence 2011, 2012, 2013, 2014.
The results from the 2 forms of counting will always be off by 1.
i.e 10-5 = 5 and the amount of numbers 5,6,7,8,9,10 -> 6 are off by 1 
A: If you count 2011 to 2014 as intervals and get only 3 years, it's because you're excluding one of the years. The interval from midnight January 1 2011 to midnight January 1 2014 is in fact three years; but it excludes all of 2014. The period from January 1 2011, to December 31 2014 is four years. That period spans the years 2011, 2012, 2013 and 2014.
You're mixing up counting and measuring.
The issue doesn't change when you measure from May to May; that is just an offset. From May 2013 to May 2014 is a period of one year, which involves two calendar years.
A: This is the problem of counting intervals lengths vs. counting markers separating them:
$$
|_{11}\;X\quad|_{12}\;X\quad|_{13}\;X\quad|_{14}
$$
Let's assume for simplicity that we're in May 2014.
If you want to know the length of the time you worked, sum the intervals (the $X$'s).
If you want to know how many different years you've been in during your work, count the delimiters (the $|$'s)
A: I got into a small argument with a friend of mine over this same problem in elementary school. When reading a book, my friend would say that he's read, say, 10 pages once he gets to page 10. I would say that he's only read 9, because he hasn't actually read page 10 if he's just gotten there. You've only worked 3 years (give or take a couple of months). The reasoning is: You've reached, but not finished, your fourth year. Therefore, the current year that you're working shouldn't be counted. Whenever you count the years, just count up to the year before the current year. So we count 2011, 2012, and 2013, but not 2014 because 2014 isn't finished yet.
A: You've worked in 4 years. In this sense, a "year" is a numbered calendar year like 2011, 2012, 2013, or 2014. Whether one has worked in a particular numbered calendar year is a yes-or-no matter of whether one has worked on any day in that year.
On the other hand, you've worked for 3 years. In this sense, a "year" is not a numbered calendar year but rather a particular length of time (365 and/or 366 days).
A: You have worked three full years (2011,2012 and 2013). You are currently half-way through 2014.After this year has gone by you should also count the interval 2014-2015.
A: I used to face same kind of confusion, earlier. Then I realized that it becomes very easy if I term the day I started my job as a work anniversary day and center my calculation on that day. and start by counting the years,then months and so on.
For eg. I started on 27th July 2009.(and honestly I did!)
By today, I had worked for 5 years $2014-2009=5$ and 1 day.
Assuming you started on 12th May of 2011, By 2014 you have worked for 3 years $2014-2011$ and $7-5=2$ months and $28-12=16$ days.
So your overall tenure : $3 years$,$2 months$ and $16 days$
Cheers!
A: One more example. 
I have movie tickets to sell. If I sell tickets #001 thru (including) #010, I've sold 10 tickets. Simple? 
But if I sell #345 - #354, and subtract, I get 9. I need to account for the 'end' ticket and not just subtract numbers. 
