New update: new foreign relations stability calculation.
The existing rules based on temporary attitude factors was too punishing because it was too easy to accumulate negatives than it was to gain positives, especially because positives with one civilisations there often also cause negatives with others.
Now the most important factor are the attitude totals for the civilisations that are equal to you in score. Predominantly positive attitudes can give positive stability, predominantly negative attitudes can give negative stability. Having all civs neutral to you or positive and negative attitudes canceling each other out can lead to no or only minor stability changes from relations.
In addition, if civilisations more powerful than the group with similar score to yours predominantly like you, you get additional positive stability. Conversely, if civilisations less powerful than the group with similar score to yours predominantly dislike you, you get additinal negative stability. This makes it easier for lower ranked civs to gain stability from their foreign relations and makes it more important for powerful civs not to completely disregard the opinions of less powerful civs.
All in all however, it should be very difficult to accrue negative relation stability in the range that has previously been reported, and it is now possible to get positive stability from foreign relations. In general, the impact of foreign relations is supposed to be limited and not a deciding factor for your overall stability.