As Xie Wei mentioned, the term “measure word” / 量词 is actually obscuring a bit of complexity here. The quick and dirty version is that there are 量词 that are “count-classifiers” — say something like 条 in 一条鱼 or 一条狗, 只 In 一只貓, 一只狗, etc., or whatever. These tell you what category of thing the noun following the classifier belongs to. Then there are 量词 that give you information about the quantity of the noun, like 杯 in 一杯茶 or 堆 in 一大堆麻烦 or 些 in 这些王八蛋.
Wikipedia’s article on this (Chinese classifier) is pretty good; see in particular the section on “Count-classifiers and mass-classifiers.” There are better run-downs out there, though; Y.R. Chao’s A Grammar of Spoken Chinese is probably the most readable. Chao draws finer distinctions than I’m doing here and if you want to know more about classifiers you should get a copy and turn to page 584. He describes eight kinds of what he calls “numeratives,” but I think for our purposes it will probably be okay if we just talk about Wikipedia’s “count classifiers” and “mass classifiers.”
So basically: 个 can substitute for count classifiers, or 量词 that are talking about discrete or quantifiable items — there’s actually a term for this, “量词个化” — but not for mass classifiers, where you’re talking about something that doesn’t come in neat, discrete units: you can go into a restaurant and order “三个菜” (“three dishes,” where 菜/”dish” is a discrete unit), but not “一个茶” (“a tea,” where 茶/”tea” is a mass noun rather than a count noun). Switching up your 量词 will give you different meanings: consider the difference between 一杯茶 and 一壶茶 — both describing quantities of tea — and 一种茶, which is talking about a kind of tea.
Also, some words are more amenable than others to 个 as a count classifier: people will substitute 个 for 张 in speech when they’re talking about, say, desks (一个桌子, rather than the more “proper”一张桌子 — and people do this sort of thing all the time, whether they admit it or not), but as others have mentioned above, something like “一个纸” just sounds bizarre. (Specifically, it sounds — to me, at least — like you were using 纸 as an adjective, but then forgot to include the noun after it — saying something like “一个纸老虎” is perfectly grammatical.)
So I guess that’s by way of answering the question of whether or not 个 could replace all measure words: sure, theoretically, in the case of a certain category of measure words. And in fact, despite what some people have said in the answers here about the “original beauty” of the language, most native speakers of Mandarin are probably not aware of how frequently they use 个 instead of the more “correct” measure word when they speak. (You can see this in some of the other answers in this thread.)
Then there’s the question of whether or not it will replace other words. Much harder to answer, but I think there are probably two forces acting in opposition to each other here. One is the fact that most people are not native speakers of standard Mandarin, and so will not necessarily use the “correct” classifiers when they speak. 个 is as useful a fallback for non-native Chinese speakers of Mandarin as it is for foreign ones, and there has been a definite trend throughout the development of Mandarin towards simplification on this front.
On the other hand, there are standardizing forces standing against this, like increased literacy and broadcast media that are officially required to promulgate “correct” usage. Standard Mandarin is increasingly becoming a part of people’s lives from an early age, regardless of what their native language may be, so people will probably have less and less need of a fallback.
tl;dr: No, except sort of in some places maybe. Measure words are nowhere near as timeless and immutable as they may appear, and there is a historical trend toward simplification in Mandarin. In speech, rather than in writing, people often use 个 instead of something else, but 个 is not capable of replacing 量词 that indicate the quantity of a mass noun, and there are some classifier/noun pairings that are so deeply imprinted upon people that they will probably not absent-mindedly substitute 个.