Edited 2 January 2011 to clarify:
I don't know if there is an established meaning for general position in this context: it would be nice to have a clarification of terminology (as we've seen from the comments discussion). One possible definition is that a set of subspaces is in topological general position if for any small perturbation, there is a homeomorphism taking one configuration to the other. For 2-planes in $\mathbb R^4$, this just means that any two intersect in a point. If you intersect the subspaces with a small sphere about the origin, this gives a great circle link in $S^3$; for any $n$, there is a finite set of possible link types for these links.
Example for topological general position
For topological general position, there is a relatively simple example satisfying the question: take $n$ complex 1-dimensional subspaces of $\mathbb C^2$. The intersection with $S^3$ gives $n$ circles in a Hopf fibration. Now modify the Euclidean metric of $\mathbb C^2$ by
changing the metrics of concentric spheres to homogeneous metrics on $S^3$ obtained by
keeping the Hopf fibers rigid, but uniformly expanding their orthogonal planes by an increasing function of radius $r$. These metrics are invariant by $U(2)$, with nonpositive curvature. (This behaviour is closely related to the geometry of
$\mathbb{CP}^2$ and of complex hyperbolic space $\mathbb {CH}^2$: the family of concentric spheres in either case has this same family of metrics on $S^3$ up to a constant that depends on $r$).
Possible stronger requirements for general position
At first it may be tempting to strengthen the condition of topological general position to require that for any small enough perturbation there is a linear map carrying one pattern to another, but this condition is unreasonably strong. The linear equivalence condition is satisfied for almost every triple of 2-dimensional subspaces of a 4-dimensional vector space, since two of the subspaces can be transformed into orthogonal coordinate planes, whereupon the third subspace is the graph of a map from one to the other; by a linear transformation, this can be arranged to map basis vectors to basis vectors. When there is a fourth subspace, the map associated with the third followed by the inverse of the map associated with the fourth is a linear map from a plane to itself, and its characteristic polynomial is an invariant that varies in any open neighborhood of configurations. Note, however, that there are open sets of quadruples of 2-dimensional subspaces for which this self-map has a pair of complex conjugate eigenvalues; when this happens, the quadruple of subspaces is linearly equivalent to a quadruple of complex 1-dimensional subspaces of $\mathbb C^2$.
There are many possible intermediate conditions that could be imposed, depending on what set of special situations you focus on. If you consider all
linear maps defined by triples of planes, as above, you might want to require that the rank of any composition is constant in a neighborhood. However, for many (most) configurations of $n$ planes, when $n$ is high enough, these generate a dense set of linear maps among the planes, and small perturbations can change the rank of something. A more plausible requirement is that the dimension of the Zariski closure of the set of all compositions
of these maps
is constant, or perhaps that the dimension of the Zariski closure of the set of all compositions associated with triples in a subset remains constant. But this definition is special to the setting of half-dimensional subspaces of an even-dimensional space so it's not ideal. It's not hard to modify this for a more general setting, but there are too many possible choices: it's not self-evident that there is one `right' definiton.
One way around the issue is to rephrase the question: is there an open set, or a set of positive measure, or a generic set (intersection of open dense sets)
of $n$ 2-dimensional subspaces of $\mathbb R^4$, such that ___ [a given property is true].
Examples for an open set of subspaces
Now make a random $C^2$-small perturbation of the Hopf fibration to be a foliation of $S^3$ by geodesics, so that the perpendicular plane field $\tau$ is still a contact plane field.
We can construct new metrics in a neighborhood of the origin in $\mathbb R^4$ by modifying
only in the 2-planes orthogonal to the family of 2-planes that are cones on these geodesics, scaling these planes by multiplication by a function $f(r)$ that is concave upward (to aim for negative curvature). If $f$ has $C^\infty$ contact to the constant function 1 at $r = 0$, the resulting metric is smooth. It is flat in each plane that is a cone on leaf of the foliation.
I think that for appropriately chosen $f$ these metrics have nonpositive sectional curvature, but I'll need to come back to this later to back it up (or tear it down), unless someone else will do it.
Best Answer
Once we considered a similar problem but around infinity, try to look in our paper "Asymptotical flatness and cone structure at infinity".
Let us denote by $r$ the distance to the singular point. If dimensions $\not= 4$ then the same method shows that at singular point we have Euclidean tangent cone even if curvature is "much less" than $r^{-2}$ (say if $K=O(\tfrac{1}{r^{2-\varepsilon}})$ for some $\varepsilon>0$, but one can make it bit weaker).
In dimension 4 there might be some funny examples: Your singular point has tangent space $\mathbb R^3$, the $r$-spheres around this point are Berger spheres, so its curvature is very much like curvature of $r\cdot(S^2\times \mathbb R)$, the size of Hopf fibers goes to $0$ very fast. However if you know that dimension of the tangent space is $4$ then it has to be Euclidean.
All this can happen if curvature grows slowly. If it is bounded then one can extend Bishop--Gromov type inequality for balls around singular point. It implies that the dimension of the tangent space is $4$. That will finish the proof.