The Confederate objectives for the war were to either prolong the conflict to the point that the Union got tired of fighting or have Britain or France intervene on their behalf. The first is hard to measure, but it may have been possible before the battle of Antietam in 1862. Lee's failed invasion of the North galvanized the Union and made abolishing slavery a clear objective of the war's end. Of course there was still dissent with the Union leadership and the war in general, but Antietam gave the war a better sense of purpose.

Foreign intervention may have been the CSA's best bet. By relying on their dominance in the cotton trade, the Confederates hoped to force Great Britain and France to provide military or diplomatic support. Napoleon III was pretty openly supportive of the Confederacy, but was hesitant to provide help without British consent. In Britain, the debate was much more heated. On one hand, they relied heavily on Southern cotton for textile production, but on the other, many did not want to fight a war to defend slavery- which the war increasingly became. Additionally, losing the United States as an ally was a major concern for Britain.

One high point for European support of the CSA was after their victory at Seven Pines, which decisively pushed the Federal forces away from Richmond. Parliament debated the matter, but the Prime Minister ultimately decided to wait for more victories before intervention. As it turned out, not enough decisive victories came through for the CSA, and intervention never happened on a meaningful scale. One notable exception was the construction of ships for the Confederate navy, one of which was the famed CSS Alabama, a screw sloop that destroyed the New England whaling fleet and generally caused havoc for the Union until she was sunk by the USS Kearsarge off of Cherbourg in 1864.

Edit: So to actually answer your question, I do think that it was possible for the CSA to secure independence, provided they gained foreign support or won enough early victories. Holding important strategic points, such as the Mississippi River and coastal ports, would have helped as well.

Sources:

McPherson's "Antietam" Sven Beckert's "Empire of Cotton" is also a great resource for the war's impact on the cotton industry.

There a lot of history-what-if involved in this kind of a question, so I'll try to break it down into, say, 2 camps: military and political. I do this because if you're going to examine the possibility of a Confederate victory, one or both of these considerations has to be taken into account.

First, the obvious: militarily. By this I mean, could the Confederacy have affected a military victory over the Union at any point during the war? Short answer: no, not really. To claim a military victory, the Confederacy would have had to physically topple the Union government (so occupy Washington, D.C.) and disrupt the U.S. government's ability to prosecute the war via Congress and/or the supply infrastructure. To accomplish this, the Confederacy would have had to have an occupying force capable of a sustained presence in places like Washington, D.C., New York City, and maybe even Philadelphia so that they might be able to disrupt the Union's ability to continue the war. This wasn't the kind of military Jefferson Davis and the Confederacy had, however. For most of the war, when the South was on the offensive, they were moving to gain supplies and invade portions of the Union to prosecute a political objective. This objective was to impress England and France, and gain foreign recognition (more on this in a moment). So from a military standpoint, while the Confederacy was a very capable fighting force (both in the east and the west), they simply weren't built (or supplied and equipped in such a way) to prosecute a war of conquest. Washington, D.C. was extremely well fortified, and Lincoln was always mindful of protecting the Union's home-bases(s), if you will, so as to protect against a striking political victory.

Even when Lee and the Confederacy were at their best (think just after Chancellorsville), D.C. was never in any real danger, and the U.S. government and infrastructure remained untouched. What's more, the Confederacy was never able to effectively cooperate between their Eastern and Western armies in a way that brought the full might of their forces into concert. A great example of this was Johnston's inability to relieve Pemberton at Vicksburg, and and feuding that took place within Bragg's camp in 1863. In short, the Confederacy was never able to raise the kind of army that could seriously contend with the manufacturing and manpower capabilities of the Union. The Confederacy had some amazing generals that were able to put together a number of spectacular victories, but Lincoln knew how to keep the political and military machinery of the Union running despite all the setbacks. The Confederacy simply didn't have the railroads, the raw materials, or the manpower to compete in a traditional man-for-man struggle.

But that was never really in question. Lee and Davis knew that if the Confederacy was ever to have any chance of independence, it would have to be via foreign recognition. In this regard, the CSA's diplomatic efforts fared even more poorly than their military's. CSA diplomat John Slidell got nowhere in France in his efforts to get so much as an audience with anyone important, and his U.K. counterpart, J.M. Mason did little better. Although some historians have argued over the significance of the Emancipation Proclamation in influencing European opinions on whether to support the Confederacy, most seem to agree that it did indeed play some role in shutting the door on any slim hopes that the Confederacy might gain British or French recognition.

So, TD;LR - The Confederacy never had any real hope of beating the Union in a straightforward military engagement (and they knew it), and although Lee and the Confederacy did put together some amazing campaigns that certainly knocked the Union for a few loops, foreign intervention never came close to materializing.

[Sources - James McPherson, "Battle Cry of Freedom"; Bruce Catton, "Grant Moves South" & "A Stillness at Appomattox"; Doris Kearns Goodwin, "Team of Rivals"]