Recently, correlation analyses between Galactic dust emission templates and a number of cosmic microwave background data sets have led to differing claims on the origin of the Galactic contamination at low frequencies. In 1999, de Oliviera-Costa et al. presented work based on Tenerife data supporting the spinning dust hypothesis. Since the frequency coverage of these data is well suited to discriminate spectrally between spinning dust and free-free emission, we used the latest version of the Tenerife data, which have lower systematic uncertainty, to study the correlation in more detail. We found, however, that the evidence in favour of spinning dust originates from a small region at low Galactic latitude where the significance of the correlation itself is low and is compromised by systematic effects in the Galactic plane signal. This signal is not necessarily characteristic of dust structure, since a simple geometric model for the Galactic plane fits the data as well as the templates. The rest of the region was found to be uncorrelated. Regions that correlate with higher significance tend to have a steeper spectrum, as is expected for free-free emission. Averaging over all correlated regions yields dust correlation coefficients of 166 +/- 43 and 114 +/- 16 muK (MJy/sr)(-1) at 10 and 15 GHz respectively. These numbers, however, have large systematic uncertainties that we have identified, and care should be taken when comparing them with results from other experiments. We do find evidence for synchrotron emission with spectral index steepening from radio to microwave frequencies, but we cannot make conclusive claims about the origin of the dust-correlated component based on the spectral index estimates. Data with higher sensitivity are required to decide about the significance of the dust correlation at high Galactic latitudes, and other Galactic templates, in particular H alpha maps, will be necessary for constraining its origin.