Here's a quote from the wikipedia article on the empire
The main cause was a failure of leadership, as Lee argues the first 10 sultans from 1292 to 1566, with one exception, had done quite well. The next 13 sultans from 1566 to 1703, with two exceptions, were lackadaisical or incompetent rulers, says Lee. In a highly centralized system, the failure at the center proved fatal.
Is this a fair assessment, or were there external pressures outside of the sultan's control?
I've heard that hypothesis before too, and there may be some truth to that. Succession format within the empire did change in the period of the early 17th century though, which is typically suggested as evidence of a sea change in the quality of the office of Sultan.
Prior to Ahmed I (r. 1606-1617), a Sultan's sons would rule a province while their father still lived. Upon his death, they would compete against one another, with one winner becoming the Sultan and the rest becoming dead pretenders.
When Ahmed I ascended, he allowed two of his brothers to live, and succession was hereafter handed to the oldest male in the dynastic line of succession. So rather than a Sultan's son taking over after his death, the office could be (and several times was) passed to a brother as opposed to the deceased Sultan's eldest/surviving son.
Virginia Aksan's "Ottoman Wars: 1700-1870 An Empire Beseiged" presents a different argument though. Competition with the Hapsburg and Romanov dynasties of Austria-Hungary and Russia respectively took their toll, to be sure. Of these three rival imperialistic empires, multi cultural and multi-ethnic in their make-up, none survived WWI intact.
They faced similar challenges during the period of modernization and industrialization, social (e.g. how to get people in the empire to view themselves as Ottoman citizens as opposed to a more localized identity), military (e.g. how to enforce conscription, needed to field a large imperial army) and economic (how to deal with Western European expansion and growth as they carved up the Americas).
So the external pressures were myriad, and the Ottomans (like their Hapsburg and Romanov counterparts) did their best to adapt, but ultimately failed. So my interpretation would be that it is not fair to blame the fall of the Ottoman Empire on a series of weak Sultans. That would discount many external factors that colluded together with a great variety of internal pressures.
Edit: mislabeled my century :( Changed mention of 'early 16th century' to 'early 17th century' in paragraph one.
As a general rule of thumb, anything that sounds like Great Man History (which is basically the idea that history is primarily shaped by Great Men) is going to get you into trouble. It's an enormous over-simplification. And while there are some cases where you can probably argue that it's accurate, it's just generally not going to cut it.
And the Ottoman Empire is a great example of how it just doesn't cut it, especially later on. By the time you to get the Koprulu era (c. 1650 - c. 1700), the state is basically run by the grand vizier, not the sultan. Sultans were no longer even going on campaign (and, of course, sultans like Mehmet II and Suleyman I spent a considerable amount of their lives commanding these campaigns; their later successors didn't). I talked a bit about the Battle of St Gotthard here; that was one of the many campaigns led by one of the Koprulu family.
As I see it, the Ottomans fell for a number of reasons but these are probably the four most notable.
When Portugal sailed around Africa in the early 16th century, it set into motion a chain of events that would effectively break the traditional Ottoman (or, at the time, Mamluk-Ottoman) control over the eastern trade routes. This was such a big deal at the time that you saw a coalition between the Ottomans, Venetians, Mamluks, and some Indian states attempt to oppose the Portuguese as they exerted greater control over the region. The coalition failed. Over time, Portugal began to effectively siphon the trade away from places like Constantinople and Alexandria and towards places like Lisbon. This ultimately deprived the Ottoman Empire of a great deal of wealth and influence.
The rise of Russia. Throughout the first half of the Ottoman Empire's history, the area north of Crimea (effectively an Ottoman subject) may as well have been a black hole; Crimea and her neighbors would regularly raid it, but the nations there posed little enough threat to Crimea, nevermind the Ottoman Empire. But by the time we get to the middle of the 17th century, especially after acquiring "left-bank" Ukraine, it's become clear that Russia is a major player on the world stage. The Ottomans now had yet another threat to deal with -- and it was on a front they could traditionally ignore.
Christian coalitions. Individually, the Ottoman Empire could reasonably expect to fight just about any one state to a stand-still (except Russia; the Ottomans were terrible at fighting Russia). But the Europeans didn't just line up and come after the Ottomans one at a time: they formed coalitions. Whenever the Ottomans seemed to be doing unusually well, Christian states (usually some combination of the Papacy, Spain, Austria, Venice, Genoa, Poland, and France) would band together and check the Ottoman advance. A pretty good example of this is Lepanto, which basically pitted the Ottoman Empire against almost every other Mediterranean state. Lepanto was the culmination of a century-long drama involving piracy, Khayr-ad-Din, a short-lived Franco-Ottoman alliance, and an unprecedented string of Ottoman naval successes prior to all that. Similar situations elsewhere (like the 'Great Turkish War') would crop up periodically through history. And while it's true that Christian coalitions didn't always work (the debacle of Varna underscores that point pretty well), they presented an inherent check on Ottoman expansion.
Nationalist awakenings. For the most part, the Ottoman Empire didn't really care who you were or what faith you followed. As long as you paid your taxes, the government didn't really care what faith you followed (note how Mehmet II kept the Greek Orthodox Church around after capturing Constantinople). Ottoman efforts at conversion were slow and usually community-level projects. But without a common religion to unify the empire, Ottoman subjects -- especially in the Balkans -- started clamoring for more autonomy and, in some cases, outright independence. The net result of this is that the Ottomans were obliged to deal with simmering discontent at around the same time they were in desperate need of military modernization. And, of course, the Europeans got involved whenever there was a big conflagration, like when Greece declared independence.
It's really hard to boil down "why did this centuries-old empire fall?" to something short and sweet. But those are, at least in my view, the four biggest reasons the Ottoman Empire fell. I could certainly expand the list and talk about things like Ottoman military performance in the 19th century, the distinct blend of incompetence and malevolence that was Enver Pasha, poor prioritization of military objectives in the 19th and 20th centuries, and so forth. But the truth of the matter is that the Ottoman Empire was already doomed by the 19th century. It would have taken a miracle to salvage the empire at that point and that miracle, of course, didn't happen.
Edit: I think I got every vowel in malevolence wrong. That was embarrassing.
If I'm remembering correctly, Niall Ferguson argued that the Ottoman Empire was hamstrung by anti-science/anti-entrepreneur/anti-inovation problems.
They went from being world leaders in astronomy and trade relationships to supporting a form of isolationism defined by religious doctrine/dogma.