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I think what you are describing is when you hang a weight on a string that is supported at both ends. The weight pulls the string down and some angle is formed between the original horizontal string and the new string position.
This is a matter of recognizing that the string, in supporting the weight, has two vector components involved in the support.
If you draw a line vertically up from where the weight is suspended on the string, you will have two right triangles. So in effect, the left half of the string is both countering half of the weight by exerting a force in the y-direction, and also exerting a force in the x-direction to keep weight from moving to the right. Likewise, the right half of the string is exerting a force in the y-direction equal to half the weight, and a force in the x-direction equal and opposite to the force of the left string in the x-direction.
Now consider what happens as the angle between where the string is attached gets larger. As that angle increases, more of the vector force of the string is directed in the y-direction and it can then support a larger weight. Likewise, as the angle decreases, more of the vector force of the string is in the x-direction and less in the y-direction so the string can support less weight.
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