death.  In reality, his words are quite true.  MeetingJuliet at the party sets into motion the events that will cause him to take his own life several days later.

After Romeo lowers himself from Juliet's balcony and is standing in the orchard below, he promises Juliet that he will see her again. Juliet responds by offering Romeo an ominous premonition. Juliet informs Romeo that she has an "ill-divining soul" before saying,

"Methinks I see thee now, thou art so low As one dead in the bottom of a tomb. Either my eyesight fails, or thou look’st pale." (Shakespeare, 3.5.54-56)

Romeo replies by telling Juliet that she also looks unusually pale before he leaves for Mantua. Tragically, Juliet's premonition is accurate and comes to fruition later in the play. After Juliet's parents inform her of the impending wedding to Paris, Juliet consults Friar Laurence, who gives her a strong sleeping potion that will make it seem like she is dead for forty-two hours. Unfortunately, Friar Laurence is unable to relay the message to Romeo, who actually believes the reports that Juliet is dead. When Romeo arrives at Juliet's tomb, he mistakenly believes that Juliet is dead and commits suicide. When Juliet wakes up, she sees Romeo's dead body and also commits suicide.

In Act 5, scene 1, Romeo declares the following:

"If I may trust the flattering truth of sleep,
My dreams presage some joyful news at hand:
My bosom's lord sits lightly in his throne;
And all this day an unaccustom'd spirit
Lifts me above the ground with cheerful thoughts.
I dreamt my lady came and found me dead--Strange dream, that gives a dead man leaveto think!--
And breathed such life with kisses in my lips,That I revived, and was an emperor.
Ah me! how sweet is love itself possess'd,
When but love's shadows are so rich in joy!"

This is clearly a premonition that he has had. The irony is that the reality does not have the joyful ending predicted in the dream. The dream has made Romeo light in spirit - it has pleased him and put him in a less morose mood. He is now in a different frame of mind. The dream is so unusual that it would drive even a dead man to think! However, even though Juliet does in reality find him dead later when she awakens from her death-like slumber, her kisses do not revive him and she, instead, commits suicide.

Tragically, it is immediately after this declaration that Romeo is informed by Balthasar about Juliet's 'death' and her burial. He then plots his own demise by purchasing an extremely powerful poison from an apothecary. 

Earlier in the play, In Act 1, scene 4, Romeo does express some misgivings about attending the Capulet ball and, although he accurately predicts the events to follow, this is not, in the true sense of the word, a premonition. Romeo is merely expressing a feeling or thought as one would when one is about to take a risk, as they were at the time, by venturing to an occasion hosted by their enemies.  

Romeo's premonition comes at the end of Act I, scene iv, right before he and his friends go off to Capulet's party.  Here is the text:

• I fear, too early: for my mind misgives
  Some consequence, yet hanging in the stars,
  Shall bitterly begin his fearful date
  With this night's revels; and expire the term
  Of a despised life, clos'd in my breast,
  By some vile forfeit of untimely death:
  But He that hath the steerage of my course
  Direct my sail!

He imagines that this night's party is about to start some turn of events that will result in death.  However, he also feels that he has no control over these events - fate "hath the steerage of [his] course."  Of course, as readers we have already been told that there will be deaths of some "star-cross'd lovers".  Shakespeare told us in the prologue.  Here, from Romeo's own mouth, is a reminder!

Before Romeo, Benvolio, and Mercutio attend Lord Capulet's ball, they discuss how best to explain their presence, and Benvolio suggests that they try their best to maintain a low profile. When Benvolio mentions that they will simply dance and leave, Romeo responds by saying that he refuses to dance and may as well hold the torch. Romeo is still lovesick and has no desire to attend the ball or entertain the thought of being with another woman other than Rosaline. Mercutio proceeds to make sexual jokes at Romeo's expense before Romeo brings up a premonition he received in a recent dream. Romeo goes on to say that he feels they should not attend Lord Capulet's ball before Mercutio interrupts him by giving his famous speech about Queen Mab. Mercutio then rambles on about a tiny fairy queen before Romeo stops him. Once Mercutio finishes his entertaining speech, Romeo elaborates on his premonition by saying,

I fear too early, for my mind misgives
Some consequence, yet hanging in the stars,
Shall bitterly begin his fearful date
With this night's revels, and expire the term
Of a despised life closed in my breast,
By some vile forfeit of untimely death.
(Shakespeare, 1.4.104–109)

Essentially, Romeo is saying that he feels as if something bad will begin tonight if they attend the ball, which will ultimately lead to his untimely death. Despite his premonition, Romeo declares that his life is in the hands of destiny and leads his friends to the Capulet estate. Romeo's premonition foreshadows the unfortunate events that will transpire later in the play. Romeo's presence at the Capulet ball leads to his secret marriage with Juliet and incites Tybalt's rage, which has significant consequences and indirectly leads to Romeo’s death.