In the country there're $N$ cities and some pairs of cities are connected by two-way airlines (each pair with no more than one). Every airline belongs to one of $k$ companies. It turns out that it's possible to get to any city from any other, but it fails when we delete all airlines belonging to any one of the companies. What is the maximum possible number of airlines in the country ?
1. **Graph Construction**:
   - Let the cities be \( c_1, c_2, \ldots, c_N \).
   - Construct a graph \( G \) with \( N \) vertices \( v_1, v_2, \ldots, v_N \).
   - There is an edge between \( v_i \) and \( v_j \) if and only if there is a flight service between cities \( c_i \) and \( c_j \).

2. **Graph Properties**:
   - The graph \( G \) is connected because it is possible to get from any city to any other city.
   - Each edge in \( G \) is colored with one of \( k \) colors, representing the \( k \) different airline companies.

3. **Tree Subgraph**:
   - Since \( G \) is connected, there exists a spanning tree \( T \) of \( G \).
   - A spanning tree \( T \) of \( G \) has \( N-1 \) edges.

4. **Coloring and Connectivity**:
   - The number of different colors (airline companies) used in \( T \) is at most \( N-1 \) because \( T \) has \( N-1 \) edges.
   - If we remove all edges of any one color, the graph \( G \) becomes disconnected. This implies that each color must appear in the spanning tree \( T \).

5. **Maximum Number of Airlines**:
   - To maximize the number of airlines, each edge in \( T \) should be colored with a different color.
   - Therefore, the maximum number of airlines \( k \) is \( N-1 \).

6. **Sharpness of the Bound**:
   - Consider a tree \( \tau \) on \( N \) vertices with \( N-1 \) edges, where each edge is colored with a different color.
   - This tree \( \tau \) satisfies the condition that removing all edges of any one color disconnects the graph.

Thus, the maximum possible number of airlines in the country is \( N-1 \).

The final answer is \( \boxed{N-1} \).