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In versions 10+, @Daniel's approach can be used more directly using RelationGraph with the relation IntersectingQ (thanks: Carl Woll). ConnectedComponents of the resulting graph gives the desired result:

data = {{1, 2}, {1, 5}, {2, 6}, {3, 7}, {4, 8}, {5, 6}, {5, 9}, {9, 13}, {10,11}, 
   {11, 12}, {11, 15}, {15, 16}};

ConnectedComponents[RelationGraph[IntersectingQ, data]] 

{{{1, 2}, {1, 5}, {2, 6}, {5, 6}, {5, 9}, {9, 13}},
{{10, 11}, {11, 12}, {11, 15}, {15, 16}},
{{4, 8}},
{{3, 7}}}

In versions 10.2+, @Daniel's approach can be used more directly using RelationGraph with the relation IntersectingQ (thanks: Carl Woll). ConnectedComponents of the resulting graph gives the desired result:

data = {{1, 2}, {1, 5}, {2, 6}, {3, 7}, {4, 8}, {5, 6}, {5, 9}, {9, 13}, {10,11}, 
   {11, 12}, {11, 15}, {15, 16}};

ConnectedComponents[RelationGraph[IntersectingQ, data]] 

{{{1, 2}, {1, 5}, {2, 6}, {5, 6}, {5, 9}, {9, 13}},
{{10, 11}, {11, 12}, {11, 15}, {15, 16}},
{{4, 8}},
{{3, 7}}}

In versions 10+, @Daniel's approach can be used more directly using RelationGraph with the relation Intersection[##] =!= {}&. ConnectedComponents of the resulting graph gives the desired result:

data = {{1, 2}, {1, 5}, {2, 6}, {3, 7}, {4, 8}, {5, 6}, {5, 9}, {9, 13}, {10,11}, 
   {11, 12}, {11, 15}, {15, 16}};

ConnectedComponents[RelationGraph[IntersectingQ, data]] (* thanks: Carl Woll*)

{{{1, 2}, {1, 5}, {2, 6}, {5, 6}, {5, 9}, {9, 13}},
{{10, 11}, {11, 12}, {11, 15}, {15, 16}},
{{4, 8}},
{{3, 7}}}

In versions 10+, @Daniel's approach can be used more directly using RelationGraph with the relation IntersectingQ (thanks: Carl Woll). ConnectedComponents of the resulting graph gives the desired result:

data = {{1, 2}, {1, 5}, {2, 6}, {3, 7}, {4, 8}, {5, 6}, {5, 9}, {9, 13}, {10,11}, 
   {11, 12}, {11, 15}, {15, 16}};

ConnectedComponents[RelationGraph[IntersectingQ, data]] 

{{{1, 2}, {1, 5}, {2, 6}, {5, 6}, {5, 9}, {9, 13}},
{{10, 11}, {11, 12}, {11, 15}, {15, 16}},
{{4, 8}},
{{3, 7}}}

In versions 10+, @Daniel's approach can be used more directly using RelationGraph with the relation Intersection[##] =!= {}&. ConnectedComponents of the resulting graph gives the desired result:

data = {{1, 2}, {1, 5}, {2, 6}, {3, 7}, {4, 8}, {5, 6}, {5, 9}, {9, 13}, {10,11}, 
   {11, 12}, {11, 15}, {15, 16}};

ConnectedComponents[RelationGraph[Intersection[##] =!= {}&, data]]

{{{1, 2}, {1, 5}, {2, 6}, {5, 6}, {5, 9}, {9, 13}},
{{10, 11}, {11, 12}, {11, 15}, {15, 16}},
{{4, 8}},
{{3, 7}}}

In versions 10+, @Daniel's approach can be used more directly using RelationGraph with the relation Intersection[##] =!= {}&. ConnectedComponents of the resulting graph gives the desired result:

data = {{1, 2}, {1, 5}, {2, 6}, {3, 7}, {4, 8}, {5, 6}, {5, 9}, {9, 13}, {10,11}, 
   {11, 12}, {11, 15}, {15, 16}};

ConnectedComponents[RelationGraph[IntersectingQ, data]] (* thanks: Carl Woll*)

{{{1, 2}, {1, 5}, {2, 6}, {5, 6}, {5, 9}, {9, 13}},
{{10, 11}, {11, 12}, {11, 15}, {15, 16}},
{{4, 8}},
{{3, 7}}}