The wording in Statement 1 does not convey a clear meaning:
For every value in the list, if the value is increased by 1, the number of distinct values in the list does not change.
Here, the term in red could convey to some test-takers that EVERY VALUE in the list is to be increased by 1.
Not the intended meaning, since increasing every value by 1 will clearly have no effect on the number of distinct values in the list.
I believe that the following reflects the intended meaning of Statement 1:

25 integers are written on a board. Are there at least two consecutive integers among them?

1: If any single value in the list is increased by 1, the number of distinct values in the list does not change.

2: At least one value occurs more than once in the list.

Each of the cases tested below is composed of integers in ASCENDING ORDER.
{1, 1, 1...1, 1, 2} implies the following 25 integers:
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2}.

Statement 1: If any single value in the list is increased by 1, the number of different distinct values in the list does not change.
Test one case that also satisfies statement 2.
Case 1: {1, 1, 1...1, 1, 2}, for a total of two distinct values
Here, if 1 increases to 2, or if 2 increases to 3, the list will still contain two distinct values.
In this case, the list contains at least two consecutive integers, so the answer to the question stem is YES.

Test one case that doesn't also satisfy statement 2.
Case 2: {1, 3, 5...45, 47, 49}, for a total of 25 distinct values
Here, if any integer in the list increases by 1, the list will still contain 25 distinct values.
In this case, the list does not contain at least two consecutive integers, so the answer to the questions stem is NO.
INSUFFICIENT.

Statement 2: At least one value occurs more than once in the list
Case 1 also satisfies statement 2.
In Case 1, the list contains at least two consecutive integers, so the answer to the question stem is YES.
Case 3: {1, 1, 1...1, 1, 1}
In this case, the list does not contain at least two consecutive integers, so the answer to the questions stem is NO.
INSUFFICIENT.

Statements combined:
Case 1 satisfies both statements.

Try adjusting Case 1 so that it does NOT contain at least two consecutive integers.
Case 4: {1, 1, 1...1, 1, 3}, for a total of two distinct values
Not viable:
If one of the 1's increases to 2, then the number of distinct values will increase from two to three, violating the constraint in statement 1.
Implication:
To satisfy both statements, the list MUST contain at least two consecutive integers, so the answer to the question stem is YES.
SUFFICIENT.

The correct answer is C.