Where does General Relativity (GR) fail experimentally? Experimental shortcomings of GR? When speaking with researchers and professors at my physics department, I sometime catch this undertone in the conversation when speaking about GR: almost as if some are not entirely convinced about the general theory of relativity.
I was wondering, are there any cosmological observations which general relatively fails to predict?
As a hypothetical example: maybe some star follows an orbit that deviates from the orbit predicted by GR. 
I have taken a course on GR, so I have some basic understanding of the subject.
 A: Maybe the most famous example of potential failure of GR are rotation curves of galaxies. However, it could also be explained by additional mass (dark matter), so it's not clear yet how this will be resolved. (A similar situation happend in the early 1900s when people predicted planet Vulcan near Mercury to account for it's perhelion anomaly. We know how that turned out.)
(It is very sad that this question seem to be handled extremly emotionally, very unrationally. I've seen people screaming at each other at conferences about whether dark-matter is true or law of gravity needs to be adjusted.)
A: 
Are there any cosmological observations which general relatively fails to predict?

No.
One issue with GR is that it admits singularities (e.g., black holes, kugelblitzes, the initial singularity), a feature that most physicists think can't be correct. However, as of this date, there have been zero observations of (for example) interior of a black hole that disprove this conjecture.
On the flip side, GR has passed every test thrown at it. Almost all alternative theories of gravitation have failed one test or the other.
A: "The theory of relativity is considered to be self-consistent, is consistent with many experimental results, and serves as the basis of many successful theories like quantum electrodynamics. Therefore, fundamental criticism (like that of Herbert Dingle, Louis Essen, Petr Beckmann, Maurice Allais and Tom van Flandern) has not been taken seriously by the scientific community, and due to the lack of quality of many critical publications (found in the process of Peer review) they were rarely accepted for publication in reputable scientific journals. So, as in the 1920s, most of the critical works have been published in small publications houses, alternative journals (like "Apeiron" or "Galilean Electrodynamics"), or private websites. Consequently, where criticism of relativity has been dealt with by the scientific community, it has mostly been in historical studies.
However, this does not mean that there is no further development in modern physics. The progress of technology over time has led to extremely precise ways of testing the predictions of relativity, and so far it has successfully passed all tests (such as in particle accelerators to test special relativity, and by astronomical observations to test general relativity). In addition, in the theoretical field there is continuing research intended to unite general relativity and quantum theory. The most promising models are string theory and loop quantum gravity. Some variations of those models also predict violations of Lorentz invariance on a very small scale." (http://en.wikipedia.org/wiki/Criticism_of_the_theory_of_relativity#Status_of_criticism)
So I would say its only "shortcoming" is that it has yet to be unified with others theories, yet since it is self-consitent, that isn't really a shortcoming of GR.
A: The galaxy rotation problem is the most obvious, but the advent of computers has allowed this to be fudged (in my view) with complex distributions of dark matter. The challenge is to find a 'crucial' anomaly that can't be fudged in this way. Possibilities include: globular clusters which show a similar rotation anomaly (Scarpa et al., 2006) but are too small to be affected by dark matter, wide binary stars which also show an anomaly (Hernandez et al., 2011) (a very local case is Proxima Centauri). Other useful anomalies include the flyby anomalies. Note that these problems all occur at very low accelerations and I've suggested that GR fails under these conditions (McCulloch, 2012).
A: As far as it has been tested in  orbits of satellites and in the cosmological observations the theory of General Relativity is validated. 
Where it fails is when one pushes the  mathematical construct to the extreme, for example : point  elementary particles as in the current standard model. But this is a failure of all classical theories , and GR is a classical theory. 
For electrodynamics the conundrum of the infinity in a 1/r potential is solved by quantum mechanics, i.e. postulating probability functions instead of orbits for the atomic theory. The expectation is that when gravitation will be quantized the classical calculational singularities will be resolved.
The holy grail is towards a unification of all four forces, in a quantized mathematical model. At present the only candidate theories that can demonstrably allow for quantization of gravity and the embedding of the standard model of particle physics are string theories, which are at the research stage.
