I know NMR and X-ray crystallography can be used to determine the structure of a protein. Are there any other techniques that can be used to check whether a protein has native structure?
There are a host of techniques that can provide support for a folded, globular structure (more on native later). The premise is: if it looks like a duck, walks like a duck and sounds like a duck then there is a good chance that it is a duck. From what we have learned about the typical globular protein is that it is compact, contains secondary structures and folds/unfolds cooperatively. So the presence of large percentages of secondary structures is an indication of tertiary structure (of course there are some exceptions). If one looks at the CD or fluorescence signal as a function of increasing concentration of denaturant, globular proteins have a cooperative (s-shaped) response. No guarantee, but there are not so many examples of stable, folded globular proteins that are not cooperative and do not have considerable secondary structure. Here far-uv CD has the advantage as it can give a measure of secondary structure content. Fluorescence is generally restricted to sensing a change in environment (similar to near-UV CD). Other methods like small angle X-ray scattering or measurements of rotational correlation can provide rough dimensions of the protein in solution (although are often less informative and more labor intensive). No single method is really sufficient to answer the question. The more evidence you can collect, the more you can believe what you are working with is a folded, globular structure.
You mention NMR and X-ray. Here solution NMR has the advantage that valuable structural information can be obtained before the structure is determined (or protein crystalized). For example, 1H or 15N chemical shift dispersion (from random coil values) is a good indication of structure (which implies globularity and compactness). Coupling constants can be measured (without knowing the sequence specific assignments) which correlate with secondary structure. We routinely screen new constructs in this way before committing to a 3D structural study. Flexible or unstructured regions (mobile or unstructured loops, tails etc.) are quite obvious.
Does the protein have native structure? Hmmm - this is more difficult. Native generally refers to the structure in the biologically relevant (native) environment. Most globular proteins are fairly robust, but here functional assays will be needed to ensure what you are studying in vitro is an accurate representation of the situation in vivo. To further complicate matters, there are an increasing number of proteins that can have multiple stable conformations in solution. Others that are intrinsically disordered and fold in the presence of a signal or template. Demonstrating accuracy in protein structure is perhaps the most challenging task of the structural biologist.
As mentioned by Kurt and Bharath, CD would be the obvious and probably the easiest approach for answering your question.
Since you mention NMR and X-ray crystallography, I would also point out SAXS as another technique that can provide structural information and the overall shape of your protein. While SAXS will tell you if your protein or parts of it are folded, it can't provide information on discrete secondary structure elements (which CD can provide). On the other hand, CD can't provide you with an overall shape of your protein.
SAXS can tell you the overall size and shape of your protein molecule in solution and it can give you an indication of disorder (if the overall molecule or certain domains/regions are disordered). If your protein is very stable with rigid domains (as opposed to domains joined by floppy linkers) it can also give you a good low resolution structure of the whole molecule. The resolution of a SAXS structure is comparable to a high quality 3D EM structure. This is obviously not as informative as an X-ray or NMR structure, but it's good for interaction studies, showing which parts of a protein molecule are involved in forming multimers (dimers, trimers. etc...) or how it interacts with a binding partner (eg. which part of protein A binds to which part of protein B). SAXS also has the advantage that it is quick and easy and there's no need for the expensive protein labelling needed for NMR or for the expense, time and complicated procedures needed for growing protein crystals and solving X-ray structures.
A protein is a biological object that has a biological function. The first thing you have to be sure of is that your protein is pure and possesses its specific biological function. At this point you can use the far UV CD for structural information. By far UV CD you can get an idea if you're dealing with a globular protein or not. A morphology of the spectrum with a deep minimum around 200 nm, a maximum not very intense at around 185 to 190 nm and sometimes a small maximum at around 225 nm suggest IDP. While the classic morphology of globular proteins is different and very well known. Also, you can find out either information from the sequence (characteristics of the composition) that from the denaturation curves that should be S shaped. Denaturation requires enough pure protein and should be performed with GdnHCl until about 6 M and followed with FAR UV CD. The curve must be S shaped but can have very different slopes (you can check how large is the concentration range over which it is estimated the denaturation) that often indicate multiple domains or subunits. All previous suggestions are correct but if you have not clear evidence that the protein is biologically active, any test is only partially trusted, because it measures the state of the protein in the environmental conditions in which it is at that precise moment.