Minor problems
3.1. Sidelobes
A radar antenna is not a laser - some energy is radiated in other directions than the main beam. Secondary maxima are called sidelobes, and they cause echoes to be seen in wrong directions. Typically sidelobes are 20 to 40 dBZ weaker than the main lobe so only very strong echoes are a problem (hail, ships, ground clutter).
3.2. Chaff
Chaff (pieces of aluminium foil)  is frequently released by military aircraft during excersices. The chaff is seen on the radar display usually as narrow bands of high reflectivity that travel with the winds after emanating from a point source. Sometimes these chaff echoes can stretch for hundreds of  km. The bands are also very shallow , though this depends on the range from the radar and the amount of diffusion time.  Old chaff is spread wider and thus  more difficult to distinguish from actual weather.
3.3. Windmills
Windmills or windmill farms at short ranges (within a few km) are mainly a problem for reflectivity observations due to partial beam blockages. Partial blockages of (low) elevation beams cause a reduction in the quality of the surface precipitation estimates (at long ranges). The rotor blades are usually not a problem, but the rotor housing and the tower itself can cause a significant blockage. The beam blockage can be estimated rather well by considering the ratio of the intercepting area of the windmill and the local area of the radar beam. For velocity measurements windmills are only a problem when the beam blockage is total, otherwise they only may cause a single corrupted range bin.
3.4 Dry radome attenuation
Values given by manufacturer are very small. But there is very little knowledge of aging radomes. Coating (lead-free paint !) and washing have been tried. 
3.5. Attenuation by gases
(Also known as "atmospheric attenation" in the U.S. ) For weather radar wavelegths, only O2 and H2O are relevant gases. The gas attenuation is assumed to be constant, depending only on wavelegth. Extra attention must be paid when using litterature values: some books use one-way, some two-way values. 
3.6 Attenuation by clouds
Variable as clouds themselves are very variable. Attenuation through ice clouds is typically ignored. Attenuation through thin water clouds (Cu, Sc) is typically neglected as the path through cloud is short. When a radar ray is traveling trough dense snowfall or rain in an Altostratus cloud, it doe not care if the precipitation is evaporated below the beam or not. So attenuation in these clouds is considered as attenuation in precipitation and this is all right.
3.7 Insects
Insects scatter microwaves, and insect echoes are not cleaned by the Doppler filter as they move with the wind. Actually insects are useful as they provide Doppler wind soundings (VVP and VAD) from the clear air boundary layer. In warm air masses, especially sea breeze fronts, insects can be found up to 2-3 km in the North of Europe.  Even though the echoes are weak (-20 to +5 dBZ) sensitive radars do see them, so any data used as input for e.g. accumulated precipitation must be thresholded. 
3.8. SunTh
The Sun emits all kinds of electromagnetic radiation, including 5 cm microwaves. These are received and seen most often when the sun is near the horizon, ie the problem is worse in Scandinavia than elsewhere. 
3.9. Specular reflections (neighbours)
Specular reflections are mirror like reflections from flat reflecting surfaces. In a specular reflections the radiation is not scattered in all directions, but in a well-defined direction determined by the incidence angle. For radio waves water surfaces or windows/buildings may cause specular reflections. Specular reflections may cause problems for both reflectivity and velocity measurements because the direction of the radar beam is changed. For a reflection from a perpendicular surface this change of direction can be 180 degrees. This change of direction results in a erroneous projection of the radar observation. In the cause of a 180 degrees direction change, one can observe a "rain cell" moving in opposite direction and velocity data with the wrong sign. 
3.10 Forest fires, volcanic ash
Dielectric characteristics of volcanic ash are markedly different from liquid water and ice, but anyways large ash and soot particles can be detected by weather radars.
3.11 Bragg scattering from clear air
 Sharp inhomogenities in the refractive index of the atmosphere, such as occur, for example, at air mass boundaries, can result in the backscatter of radar power. This phenomenon is often called Bragg scattering. 
This is most important for radars of longer wavelength (particularly above tens of centimeters and into the meter range), especially those which are vertically pointed (generally the lower the elevation angle, the less likely it is that this phenomenon will be observed). The reason for this is that the atmosphere is primarily horizontally stratified. Thus stratified layers with very high refractive index gradients at their interfaces reflect some of the radar power directly back to a vertically pointing radar. The effect may be enhanced with altitude when the layers appear concave to the radar due to the earth’s curvature.
Air mass boundaries may produce large enough gradients of refractive index to result in radar backscatter. Americans have listed several other conditions in which similar gradients may exist: cloud and fog tops, convective boundaries, the tropopause, sheared stable layers, and thunderstorm outflow boundaries. All these conditions have a common consequence: they result in turbulence along the boundary. This turbulence is the responsible for producing the large-scale refractive index gradients which result in return of incident radar power.In general, Bragg scatter is much more prevalent for longer wavelength radar sets (especially wavelengths greater than 20 cm) and typically does not yield reflectivity greater than about -10dBZ. Its intensity is proportional to the magnitude of the refractive index gradients which exist. In Scandinavia, where wavelengths of weather radars are typically 5 cm and refractive index gradients small, it can usually be neglected.
3.12. Clouds
Non-precipitating water clouds (Stratus, Stratocumulus, Cumulus) give very weak echoes, typically less than -15 dBZ. Thus they can't be detected at all or only rarely with sensitive radars at very short ranges (usually less than 20 km). Ice clouds (Cirrus, Cirrostratus) are frequently detected as their reflectivities can be close to 0 dBZ. Most radars do detect them, so any data used as input for e.g. accumulated precipitation must be thresholded. It should be noted  additionally that Altostratus "clouds" in fact represent overhanging precipitation (see item 2.2), which is a frequent, widespread and relatively intensive problem (typically 0- 30 dBZ) compared to the other clouds. 
3.13. Aircraft in noise samples
To compensate for drifting noise and DC components, a noise measurement without echo is taken periodically and corrected for in subsequent processing. In the so far very rare case where an aircraft is hit, the noise sample must be rejected. Signal processing permitting, this can be done thresholding the departure from a longtime mean either by the noise evaluation algorithm itself or by the radar control software.
3.14 Flare echo
Three body scattering. Flare echo and is caused by the reflection of very large hail in the mid levels of the storm. The fingerlike protrusions of elevated reflectivity have been termed flare echoes or ‘‘hail spikes.’’ Three-body scattering occurs when radiation from the radar scattered toward the ground is scattered back to hydrometeors, which then scatter some of the radiation back to the radar. Important in USA, very rare in Scandinavia.