Four of the plot lines in A Midsummer Night's Dream certainly do relate. In fact, the plot lines crisscross and interact with each other.
Two of the most interactive plot lines involve the love interests of the four Athenian lovers. The fact that Hermia wants to marry Lysander while Demetrius wants to marry her as well forms one plot line. Also, the fact that Helena wants to marry Demetrius, whom she was engaged to before he began courting Hermia forms a second plot line. These two plot lines merge when Helena decides to betray Hermia, her best friend since childhood, by telling Demetrius of Hermia and Lysander's plan to escape out of Athens via the woods. Helena's singular motive for the betrayal is that it might, at least, get a simple thank you from Demetrius, as we see in her lines:
I will go tell him of fair Hermia's flight;
Then to the wood will he tomorrow night
Pursue her; and for this intelligence
If I have thanks, it is a dear expense. (I.i.251-254)
Helena's betrayal brings all four lovers into the woods, leading to further plot development and showing us how the four lovers' plot lines intertwine.
A third plot line involves the fact that Oberon feels betrayed and neglected by his wife Titania due to her erotic love for an Indian boy. Oberon's jealousy of the boy leads Oberon to decide on using the magical flower to distract his wife from the boy by making her fall in love with something grotesque. The initiation of the magical "love-in-idleness flower" makes plot lines intertwine even further. First, Puck mistakes Lysander for Demetrius, making Lysander fall in love with Helena instead of Hermia. Second, when Oberon learns of Puck's mistake, Oberon partially fixes it by enchanting Demetrius as well. Now both men are in love with the opposite woman they were in love with before, leading to quarrels among all four lovers. In addition to the "love-in-idleness flower" intertwining the two plot lines of the two Athenian couples, the flower also intertwines the two plot lines involving Oberon and Titania with the fourth plot line, the mechanicals preparing their play for performance. Bottom, in his foolishness and conceit, believes he can put on a winning play. Puck points out Bottom's foolishness by turning him into the proverbial ass. Ironically, Titania falls in love Bottom as an ass, thereby intertwining the plot lines of Oberon/Titania and the mechanicals. Later, the plot line of the mechanicals intertwines with the plot of the four Athenian lovers when the mechanicals perform their play before Duke Theseus on his wedding day, which happens to be the wedding day of all four lovers as well.
The convergence of all four plot lines serves to prove Shakespeare's themes concerning the foolishness of love and the foolishness of mankind.