©2017 ESO. A core-collapse (CC) supernova (SN) of Type IIn is dominated by the interaction of SN ejecta with the circumstellar medium (CSM). Some SNe IIn (e.g. SN 2006jd) have episodes of re-brightening ("bumps") in their light curves. We present iPTF13z, a Type IIn SN discovered on 2013 February 1 by the intermediate Palomar Transient Factory (iPTF). This SN showed at least five bumps in its declining light curve between 130 and 750 days after discovery. We analyse this peculiar behaviour and try to infer the properties of the CSM, of the SN explosion, and the nature of the progenitor star. We obtained multi-band optical photometry for over 1000 days after discovery with the P48 and P60 telescopes at Palomar Observatory. We obtained low-resolution optical spectra during the same period. We did an archival search for progenitor outbursts. We analyse the photometry and the spectra, and compare iPTF13z to other SNe IIn. In particular we derive absolute magnitudes, colours, a pseudo-bolometric light curve, and the velocities of the different components of the spectral lines. A simple analytical model is used to estimate the properties of the CSM. iPTF13z had a light curve peaking at Mrâ‰-18.3 mag. The five bumps during its decline phase had amplitudes ranging from 0.4 to 0.9 mag and durations between 20 and 120 days. The most prominent bumps appeared in all the different optical bands, when covered. The spectra of this SN showed typical SN IIn characteristics, with emission lines of Hα (with broad component FWHM ~ 103-104km s-1and narrow component FWHM ~ 102km s-1) and He i, but also with Fe ii, Ca ii, Na i D and Hβ P Cygni profiles (with velocities of ~ 103km s-1). A pre-explosion outburst was identified lasting â‰ 50 days, with Mrâ‰-15 mag around 210 days before discovery. Large, variable progenitor mass-loss rates (â‰ 0.01MâŠ(tm)yr-1) and CSM densities (â‰ 10-16g cm-3) are derived. The SN was hosted by a metal-poor dwarf galaxy at redshift z = 0.0328. We suggest that the light curve bumps of iPTF13z arose from SN ejecta interacting with denser regions in the CSM, possibly produced by the eruptions of a luminous blue variable progenitor star.