As a consequence of uniqueness and existence for ODE's we know that a ODE of nth order can have no more than n linear independent solutions. This question sums it up pretty well.
Unfortunately Qiaochu Yuan restricts itself to ODE with constant coefficients. Under what circumstances do we get a ODE with less than n linear independent solutions?
The argument using the existence and uniqueness theorem (plus Grönwall's inequality to show that existence is global) shows that there are exactly $n$ for any $n$'th order homogeneous linear differential equation on an interval where the coefficients are continuous and the leading coefficient is nonzero.
Where you can have fewer than $n$ is when the leading coefficient is $0$ at some point in the interval. Thus $x^2 y'' - 2 y = 0$ has only one linearly independent solution (namely $x^2$) defined on an interval containing $0$: the general solution is $a x^2 + b/x$ which is not defined at $x=0$ if $b \ne 0$.
A linear ODE of nth order has always $n$ linear independent solutions. This fact follows, for example, from the general theory of first order systems of linear equations.
