SMS messages can be redirected and silently intercepted using flaws in the SS7 protocol. That's why the NIST recommends not to use SMS for 2FA.
However, this attacks does not rely on intercepting the radio signal, it can be done remotely over internet. Your neighbor would not have an advantage for this attack. Also, this attack does not enable an attacker to bill SMS to your account at your mobile provider. To bill their SMS messages to your account, your SIM card is required, and that's the main purpose of the SIM card.
What your neighbor can do, however, is using an IMSI-catcher to intercept the radio communications and attempt to force your phone to use weaker or no encryption in order to decrypt your communications. You can prevent it by disabling older protocols (GSM, 2G) in your phone settings. For this attack to work, your neighbor will need to know your phone number or use multiple IMSI-catchers and to triangulate the positions of the phones at proximity and find yours using your address.
Today IMSI-catchers are mainly used to locate specific phones and their users, not to intercept their communications. This can be used for example by the police to identify suspects, or to identify participants to public protests against the local government (the legality of such use depends on the country).