There was a real hope of peace three weeks after Caesar crossed the Rubicon on 10 January 49. Caesar had proposed terms and those had been accepted and discussed as early as January 22nd, at Teanum. There, the consuls for 49, Cornelius Lentulus Crus and Claudius Marcellus, had met with Pompey and the refugee Labienus, who had been Caesar's right-hand-man in Gaul and had defected to the side of the boni, the conservative senatorial bloc in opposition to Caesar. From Cicero's (admittedly scattered and second- or third-hand) report we learn that they had apparently accepted Caesar's terms, at least provisionally, and that those terms were, loosely, that Caesar would come to Rome to stand as a candidate for the consulship of 48, and Pompey would show good faith by going to Spain, where he was the on-going governor for another few years. It isn't clear to me (or maybe anyone?) whether Caesar had offered to disband his forces in Italia entirely, or to send them back to Gaul. In the meantime, while he awaited a reply to his offer, Caesar continued his lightning-fast advance through northern Italia, facing little to no opposition.

Cicero had been hopeful that this preliminary agreement would hold. After all, Pompey was supposed to be in Spain anyway, since it was his province, not loitering about the countryside near Rome. And Caesar had been clear from the beginning that all he really wanted was a stop-gap means of imperium to protect him from vindictive prosecution long enough to be elected consul again. Neither were impossible to accomplish. It isn't entirely clear why or how this initial deal broke down, though we hear later that the Pompeian side had demanded that Caesar withdraw from all the Italian towns he currently occupied before any deals could be made. Maybe Caesar found that to be needlessly nit-picky. Two other factors probably did more to torpedo this deal: Pompey was supremely confident in his ability to win any war, and he was egged on by Labienus, who reported that Caesar's forces were disheartened by crossing the Rubicon and were ready to lay down their arms or flip. Both of these notions were fundamentally false, and it would take the Pompeian side far far far too long to realize that the war in Italia was already lost.

So in short: the stubbornness and pettiness of the boni in attempting to push things with the proposed deal, especially given Caesar's own stubbornness, his position of strength, and his well-known pride and volatility; Pompey's overconfidence in his own forces and incorrect assessment of Caesarian legion morale; and (probably) the false reports relating to both which Labienus furnished at Teanum. By 5 February 49, Cicero writes to Atticus "I have given up hope of peace" (ad Att 7.20).

After this, some events of the early war were out of Pompey's control. First, he had to deal with Domitius, another consular with a large army and a large head. Domitius refused to follow Pompey's directions in the early weeks of the war, and things were murky as far as who had the power to order who. After losing the Po valley and Picenum, Pompey (rightly) realized that he needed to regroup in southern Italy at Luceria, thereby essentially abandoning the capital. Domitius refused and managed to get himself hemmed in and destroyed by Caesar, despite Pompey's abject pleas (see ad Att 8.12B and 8.12C in particular). This essentially ended any hopes of a Pompeian victory in the Italian theater of war in 49. Pompey's strategy now (rightly, again) shifted to regrouping in the East, where he had a base of power. He (again, rightly) realized that his forces were not as reliable as he had imagined, and that Caesar's legions were in no way suffering morale problems--quite the opposite. Caesar's legions, after years of vicious fighting in Gaul, had become perhaps the most professional and effective fighting force in Roman history up to that point.

If there was any time to re-think a peace deal, it was at this moment, as Caesar chased the Pompeians from Luceria and down the coast to Brundisium harbor, only narrowly missing them: Caesar's legions were constructing blockades in the harbor as Pompey's fleet finally sailed away to Greece in early March. Pompey's decision to move his force of 30,000 soldiers, the two consuls Lentulus and Marcellus, and the entire boni faction of the senate-in-exile to Greece prolonged the war, but it was not a decision he had made in the moment: it had been his plan for weeks, and perhaps had been in the back of his mind since mid-January. Talking on a strategic level, Pompey was still not in a terrible position: he had the rich East as a base, and could count on supplies, coin, and manpower from Greece, Asia Minor, and Syria to build a force capable of stopping Caesar's veteran legions. He also bought himself time to train them. In Spain, he had two loyal lieutenants and an army of six legions ready to harass Caesarian Gaul and perhaps push into northern Italia from Provence, when the time came. It wasn't a bad plan.

edit for more letter references, because Cicero letters are fun. Ad Att = "Letters to Atticus"; book number and letter number are referenced. All dates refer to time of letter composition, not events within the letter (which have happened in the days/weeks previous).

7.8 (25 or 26 Dec 50): "Pompey seems set on a σύγχυσιν τῆς πολιτείας (Civil War)"

7.10 (18 Jan 49): Caesar has crossed into Italy on Jan 10; Cicero calls Pompey's response "lethargic"

7.12 (22 Jan 49): Cicero muses about Pompey's options: stay to fight in Italia or abandon Rome to the wolves for a better strategic position?

7.13A (24 Jan 49) + 7.14 (25 Jan): the letters about the meeting at Teanum; the purported agreement of terms

7.15 (26 Jan 49) + ad Fam 16.12 to Tiro in Greece: Cicero has met with the consuls at Capua, in Campania, as they head south, ostensibly to recruit more soldiers (they never will); Cicero describes the terms of this supposed peace deal, including the stipulation that Caesar must remove himself from northern Italian towns; Cicero expresses his belief that this deal will work and that disaster will be averted, and that the boni will have to stomach the bitter pill of Caesar being elected consul for 48 without further incident; Labienus has officially defected to the Pompeians

7.16 (28 Jan 49): Cicero has received a set of letters from several sources with confusing and possibly conflicting news. He hears that Pompey intends to move north into Picenum in force (this will never happen; Picenum is already lost); Pompey repeats what Labienus had said: that Caesar's legions are demoralized (they are not)

7.18 + 7.19 (3 Feb 49): Cicero is certain that Caesar will take the deal; it is reported to him 3rd-hand, incorrectly, that there has been a reversal in the north at Ancona and that Caesar's forces have been driven out; he anticipates leaving for Spain with Pompey and the other boni (this will not happen); and finally, while writing 7.19, he receives a further missive which plainly states that there will be no deals and no peace, and Cicero is despondent

7.20 (5 Feb 49): "I have given up all hope of peace"

8.11A + 8.12A + 8.12B (11-15 Feb 49): Pompey writes curtly to Cicero to come to Luceria and get involved in the damned war, since Cicero is ostensibly of consular rank and a former leader of the Republic; Cicero replies with sass and excuses and does not leave for Luceria: he will in fact remain in his villa(s) in Campania for the duration of the Italian theater; meanwhile Pompey has received word that the consular Domitius intends to make a stand at Corfinium, east of Rome in the mountains, and is horrified; he tells him to come to Luceria immediately.

8.12A (18 Feb 49): Pompey patiently explains to Cicero why he will not move north to Corfinium, and how Domitius has lost them the war in Italy; he lays out his plans for Greece and the rest of the war.

8.8 (23 Feb 49): word reaches Cicero that Domitius has been destroyed at Corfinium, that Pompey is fleeing to Brundisium, and that Italia is lost to Caesar.