SOLUTION: If a_1, a_2, ..., a_19 satisfy a_1 + a_2 + a_3 + a_4 = 1, a_2 + a_3 + a_4 + a_5 = 2, a_3 + a_4 + a_5 + a_6 = 3, ... a_{16} + a_{17} + a_{18} + a_{19} = 16, a_{17}

Let's analyze the given system of equations.

We have:
1.  a_1 + a_2 + a_3 + a_4 = 1
2.  a_2 + a_3 + a_4 + a_5 = 2
3.  a_3 + a_4 + a_5 + a_6 = 3
...
16. a_16 + a_17 + a_18 + a_19 = 16
17. a_17 + a_18 + a_19 + a_1 = 17
18. a_18 + a_19 + a_1 + a_2 = 18
19. a_19 + a_1 + a_2 + a_3 = 19

Subtract equation (1) from equation (2):
    (a_2 + a_3 + a_4 + a_5) - (a_1 + a_2 + a_3 + a_4) = 2 - 1
    a_5 - a_1 = 1
    a_5 = a_1 + 1

Subtract equation (2) from equation (3):
    (a_3 + a_4 + a_5 + a_6) - (a_2 + a_3 + a_4 + a_5) = 3 - 2
    a_6 - a_2 = 1
    a_6 = a_2 + 1

In general, we have a_{n+4} = a_n + 1.

This means:
    a_5 = a_1 + 1
    a_9 = a_5 + 1 = a_1 + 2
    a_13 = a_9 + 1 = a_1 + 3
    a_17 = a_13 + 1 = a_1 + 4

Similarly,
    a_6 = a_2 + 1
    a_10 = a_6 + 1 = a_2 + 2
    a_14 = a_10 + 1 = a_2 + 3
    a_18 = a_14 + 1 = a_2 + 4

Also,
    a_7 = a_3 + 1
    a_11 = a_7 + 1 = a_3 + 2
    a_15 = a_11 + 1 = a_3 + 3
    a_19 = a_15 + 1 = a_3 + 4

And,
    a_8 = a_4 + 1
    a_12 = a_8 + 1 = a_4 + 2
    a_16 = a_12 + 1 = a_4 + 3

Now let's use equations 1, 17, 18, and 19.

From eq. 1  -->  a_1 + a_2 + a_3 + a_4 = 1
From eq.17  -->  a_17 + a_18 + a_19 + a_1 = 17
From eq.18  -->  a_18 + a_19 + a_1 + a_2 = 18
From eq.19 -->   a_19 + a_1 + a_2 + a_3 = 19      (*)

Substituting the expressions we found:
17. (a_1 + 4) + (a_2 + 4) + (a_3 + 4) + a_1 = 17
    2a_1 + a_2 + a_3 + 12 = 17
    2a_1 + a_2 + a_3 = 5                           (17)

18. (a_2 + 4) + (a_3 + 4) + a_1 + a_2 = 18
    a_1 + 2a_2 + a_3 + 8 = 18
    a_1 + 2a_2 + a_3 = 10                          (18)

19. (a_3 + 4) + a_1 + a_2 + a_3 = 19
    a_1 + a_2 + 2a_3 + 4 = 19
    a_1 + a_2 + 2a_3 = 15                          (19) 

                                  <<<---===  till this point, the solution by @CPhill is correct.
                                             After this point, it is WRONG.

                                  <<<---===  So, starting from this point, my solution is going.

20.  Add equations (17), (18) and (19).  You will get

     4a_1 + 4a_2 + 4a_3 = 5 + 10 + 15 

     4a_1 + 4a_2 + 4a_3 = 30.


 Divide both sides by 4

     a_1 + a_2 + a_3 = 7.5.      (20)


Now we are on the finish line.


From equation (17) subtract equation (20).  You will get  a_1 =  5 - 7.5 = -2.5.


From equation (18) subtract equation (20).  You will get  a_2 = 10 - 7.5 = 2.5.


From equation (19) subtract equation (20).  You will get  a_3 = 15 - 7.5 = 7.5.


From equation (*), a_19 = 19 - a_1 - a_2 - a_3 = 19 - 0 - (-2.5) - 2.5 - 7.5 =  11.5


For completeness, let's determine a_1 from the very first equation in this post

    a_1 = 1 - (a_2 + a_3 + a_4) = 1 - ((-2.5) + 2.5 + 7.5) = 1 - 7.5 = -6.5.


From this point, all 19 terms  a_1, a_2, a_3, a_4, a_5, . . . , a_19 can be determined.

See this table below


    a_1    a_2    a_3    a_4    a_5    a_6    a_7    a_8    a_9    a_10   a_11    a_12    a_13    a_14    a_15    a_16    a_17    a_18    a_19  

   -2.5    2.5    7.5   -6.5   -1.5    3.5    8.5   -5.5   -0.5    4.5    9.5     -4.5    0.5     5.5     10.5    5.5     1.5     6.5     11.5


You may check that all given 19 original equations are satisfied.


Final Answer: The final answer is  a_19 = 11.5.