Example 4 Choose $k$ numbers from $1,2, \cdots, 2004$, such that among the chosen $k$ numbers, there are definitely three numbers that can form the side lengths of a triangle (the three numbers must be distinct). What is the minimum value of $k$ that satisfies this condition?
When selecting three numbers from 1 to 2004 to form the sides of a triangle, there are too many possibilities. Instead, let's approach it from the opposite direction and list all sets of three numbers that cannot form the sides of a triangle.

First, 1, 2, 3 cannot form the sides of a triangle. Adding 5, the set \(1, 2, 3, 5\) also cannot form the sides of a triangle.

Continuing this way, the added number is exactly the sum of the last two numbers, resulting in:
\[
\begin{array}{l}
1, 2, 3, 5, 8, 13, 21 ; 34, 55, 89, 144, 233, \\
377, 610, 987, 1597
\end{array}
\]

There are 16 numbers in total. In these 16 numbers, any three chosen cannot form the sides of a triangle. If we add any other number from the remaining ones, then among these 17 numbers, we can find three numbers that can form the sides of a triangle.
Therefore, the smallest value of \(k\) that satisfies the condition is 17.