Linacs for high-energy physics, as well as for industry and medicine, require accelerating structures which are compact, robust, and cost-effective. Small foot-print linacs require high-accelerating gradients. Currently, stable-operating gradients, exceeding $100\text{ }\text{ }\mathrm{MV}/\mathrm{m}$, have been demonstrated at SLAC National Accelerator Laboratory, CERN, and KEK at X-band frequencies. Recent experiments show that accelerating cavities made out of hard copper alloys achieve better high-gradient performance as compared with soft copper cavities. In the scope of a decade-long collaboration between SLAC, INFN-Frascati, and KEK on the development of innovative high-gradient structures, this particular study focuses on the technological developments directed to show the viability of novel welding techniques. Two novel X-band accelerating structures, made out of hard copper, were fabricated at INFN-Frascati by means of clamping and welding. One cavity was welded with the electron beam and the other one with the tungsten inert gas welding process. In the technological development of the construction methods of high-gradient accelerating structures, high-power testing is a critical step for the verification of their viability. Here, we present the outcome of this step---the results of the high-power rf tests of these two structures. These tests include the measurements of the breakdown rate probability used to characterize the behavior of vacuum rf breakdowns, one of the major factors limiting the operating accelerating gradients. The electron beam welded structure demonstrated accelerating gradients of $90\text{ }\text{ }\mathrm{MV}/\mathrm{m}$ at a breakdown rate of ${10}^{\ensuremath{-}3}/(\mathrm{pulse}\text{ }\mathrm{meter})$ using a shaped pulse with a 150 ns flat part. Nevertheless, it did not achieve its ultimate performance because of arcing in the mode launcher power coupler. On the other hand, the tungsten inert gas welded structure reached its ultimate performance and operated at about a $150\text{ }\text{ }\mathrm{MV}/\mathrm{m}$ gradient at a breakdown rate of ${10}^{\ensuremath{-}3}/(\mathrm{pulse}\text{ }\mathrm{meter})$ using a shaped pulse with a 150 ns flat part. The results of both experiments show that welding, a robust, and low-cost alternative to brazing or diffusion bonding, is viable for high-gradient operation. This approach enables the construction of multicell standing and traveling-wave accelerating structures.